INCLUDITO SQUAEE BOOT, CUBE BOOT, AND OTHER BOOTS.
A HIGHLY PRACTICAL, BRIEF AND UNIQUE METHOD
FOR THE EXTRACTION OF ALL
ARITHMETICAL ROOTS.
A SCIENTIFIC PROCESS
NOT HERETOFORE PRESENTED IN ANY PUBLISHED WORK
ON ARITHMETIC, AND
SAVING NINE-TENTHS OF THE LABOR
USUALLY NECESSARY FOR THE
EXTRACTION OF ROOTS, AND ESPECIALLY OF CUBE ROOT,
UNDER THE RULES NOW EMPLOYED.
FOR THE USE OF ALL GRADES AND ALL SCHOOLS
ABOVE THE PRIMARY, AND FOR TEACHERS
IN PARTICULAR.
^^ OF THB***^
" OF THE
lUKIVEESITY]
— BY—
G. D. HINES, A. M.
w
CLEVELAND. 0.:
J. R. HOLCOMB & CO., PUBLISHERS,
1886.
COPYBIQHT, BY J. E, HOLCOMB & Co., 1886.
,1^ -Copies of this work will be sent to any address, postpaid, on receipt of price. Active
Agents wanted. J. R- HOLCOMB & CO., Pdblishbes,
Cleveland, Ohio.
^7-9^
DEDICATION.
TO MY WIFE,
THE SYMPATHETIC SHARER WITH ME OF THE MIXED
CUP OF FORTUNE INCIDENT TO
A LONG SCHOOL LIFE, THIS LITTLE VOLUME
IS AFFECTIONATELY INSCRIBED.
G. D. HiNES.
PREFACE.
-^HIS little work needs but a short introduction. It is brevity itself, and whatever of merit it may
possess, is largely due to its brevity and practicability. The methods of treating difficult roots,
herein contained, and especially of cube root, were first suggested to the Author in the winter
of '8i and '82, while teaching the Lincoln School, Plumas County, California. The pupils of
that school, as is usual with most pupils, on first meeting the subject, were having trouble with
cube root. This caused the teacher to put his wits to work in the almost hopeless effort to
devise, if possible, some easier and shorter way of solution than the prolix processes extant in the
arithmetics ; and, after some days of close scrutiny of the meaning and relation of roots and powers,
the Author detected, for the first time, a new method of teaching cube root. The treatment of other
roots, and of surds, naturally followed ; and the result has been what my fellow-teachers and others
will find in the following pages of this work.
It is believed that the addenda of Interest, with its unique, brief, and simple treatment, will prove
an attractive feature of the book.
This work is distinctly original. It details our own discoveries and is the product of our own
thought respecting the treatment of that difficult subject, cube root. We send it forth on its mission,
conscious that it must stand or fall on its own merits. We are fully persuaded, at the same time, that
it needs only to be seen and understood to be appreciated ; and that, if generally introduced, it must
supplant the perplexing and unsatisfactory rules in the text-books on cube root and the higher roots,
and on surds. We ask an impartial examination by our fellow-teachers everywhere, and believe they
will find the little book helpful, if not indispensable, to them.
Selma, Cal., May 28, 1886.
cube"root
AND OTHER ROOTS UPROOTED.
Def. — A root is one of the equal factors that has been repeated in multiplication a given number
of times, to produce a given pov/er. Ex. — 3 is the cube root of 27, having been twice repeated in
multiplication, or thrice used as a factor, to produce the power. 27 is the cube root of 19683, having
been twice repeated in multiplication, or thrice used as a factor, to produce the power. 15 is the fourth
root of 50625, having been three times repeated in multiplication, or four times used as a factor, to
produce the power. And so on for any other roots, integral or fractional, positive or negative. To
find the roots of all powers that legitimately belong to arithmetic, is the chief object of this work.
Obs. — As this treatise deals chiefly with cube root and other higher roots, no extended notice of
square root will be taken. Only an incidental usage will be made of it.
THE BASIS.
It is a well-known fact that the principles underlying Arithmetical evolution are derived from the
mother science of Algebra, and that the arithmetical rules have been formulated out of the algebraic
formulce. No other rules for the extraction of roots have been presented in the arithmetics, and per-
haps, substantially, no others can be framed than those which depend on algebraic principles. But
certain rules can, nevertheless, be formulated, which, while they have reference to algebraic principles,
completely revolutionize the old mammoth rules, and, in brevity, almost annul them.
THE NEW TREATMENT.
The methods about to be illustrated neted have no reference to algebra, nor do they require any
knowledge of that science. They annihilate the "cubic block" system, which clearly presents the
principles of evolution to only the maturer scholars ; and then only in cube root, and are of no real
advantage in the actual work of even the cube root, in very large numbers, and certainly of no advant-
age in extracting any other than the cube root.
CUBE ROOT.
We will now present our method for the extraction of the cube root. Powers are either perfect or
imperfect. 15625 is a perfect cube, while 18740 is an imperfect cube, or third power, and is called a surd.
We will present a rule for the extraction of the cube root of perfect third powers, and one also for
that of surds.
Obs. — It may be remarked that a surd may be considered an imperfect power of any degree what-
ever. Thus, 18740 may be considered an imjaerfect square, cube, fourth power, or any other power ;
for we may require the approximate square root, cube root, fourth root, or any other root, of 18740.
But it sometimes happens that a perfect power of one degree is a surd of another degree, and vice versa.
Ex. — 25 is a perfect square, but an imperfect cube ; while 27 is an imperfect square, but a perfect
cube,
EXACT CUBES OF TWO PERIODS.
Let us extract the cube root of the following numbers, viz.:
-^ 13824
74088
185 193 A little observation and practice enable us to determine by inspection the root
250047 figure of the first period of the pow^r. Thus, the root figure of 13, in the first of
262144 91 125 the preceding numbers, is 2. And by the new method, the root figure of the last
166375 ■ 97336 period is 4, when the period ends in 4. Hence, the cube root of 13824 is 24. The
704969 226981 root figure of the first period of 74088 is 4, and the root figure of the last perlod'is 2,
when the period ends in 8, (It is 8 when the periocl ends in 2.) So the cube i-oot, of the last number
is 42. The cube root of 262144 is obtained in the same way. The root figure of 262 is 6, and of 144 it
is 4. So the ^262144 is 64. Again, the root figure of the first period of 166375 is 5; and the root
figure of the last period is 5, when the period ends in 5. So the 1^166375 is 55. 704969 gives, for the
first root figure, 8 ; and the last is 9, when the period ends in 9. So, the ^'704969 is 89. In like man-
ner, 185193 gives, for the first root figure, 5 ; and last figure is 7, when the period ends in 3. (It is 3,
when the period ends in 7.) Thus, the 1^185193 is 57. The cube root of 250047, for reasons already
MATHEMATICAL ROOTS UPROOTED.
Stated, is 63. The cube root of 9"25, for similar reasons, is 45. The ^97336 is to be written out
impromptu, just as the p revious roots have been ; making the last root figure 6, when the final period
ends in ^. Also, the ^226981 is 61, the last root figure being i, when the concluding period ends in
I. Observe that, in all perfect cubes, the final root figure is simply chosen, according to the character
of the terminating figure of the power. This is a great saving of time and work, as will be shown
hereafter.
EXPLANATION.
The reasons for the foregoing selection of the final root figure, depend on a very plain principle. In
all perfect third powers, it is evident that the final figure of the pnver arises from the multiplication
of the final figure of the root tzuice into itself. Now, if we multiply the nine digits, respectively, twice
into themselves, we shall have this result, viz.:
Observing the final figures of these powers, we see that the final root figure i pro-
duces a 1, on being twice multiplied into itself. We see that the final root figure 2
produces an 8, and an 8 a 2 ; that 3 produces a 7, and a 7 a 3 ; that a 4 produces a
4; that a 5 produces a 5 ; that a 6 produces a 6, and that a 9 produces a 9.
Obs. — The cubes of the several digits, viz.: i, 8, 27, 64, 125, 216, 343, 512, 729,
must be so thoroughly familiar to the student that he can select the root figure of the
first period impromptu.
If the first period is, in magnitude, between i and 8, the root figure is i ; if the
first period is, in magnitude, between 8 and 27, the root figure is 2 ; if the first period
is embraced between 27 and 64, the root figure is 3 ; if it is between 64 and 125, the root figure is 4;
and so on.
1. What is the root figure of the first period, if the period is comprised between 512 and 729?
2. What is the root figure of the first period, if the period is larger than 729 ?
EXAMPLES.
iXiX 1= I
2x2x2= 8
3X3X3= 27
4X4X4= 64
5x5x5 = 125
6X6X6 = 216
7 X 7 X 7 = 343
8x8x8=512
9 X 9 X 9 = 729
Let it be required to extract the cube root of the following, by the foregoing principles, viz.:
Thus, we may write out, impromptu, according to the foregoing principles, the'roots
of these, and of all perfect cubes involving only two periods.
Let the student find the roots of the following, writing out the answers, off-hand, with-
out any figuring or formal extraction, and choosing, at sight, the final root figure, ac-
cording to the character of the terminal figure of the power, viz.:
39304
#" ■091125
f 46656
f 405224
f 389017
f 474582
lA
] ^. 0000021744
^.000024389
f^ . 000079507
^ .000614125
f 2.19 7000
f^ 941. 192000
0l>s. — The last six preceding numbers consist of three periods, but may be solved without premed-
itation by the same rules, respecting the terminal figure, as are given for powers of two periods.
EXACT CUBES OF THREE PERIODS.
Let us extract the cube root of the following numbers:
: .- 7. 153990656
V'^^. I. 44361864 4- 12.812904 8. 387420489
\ Q^^ 2. 134217728 5. 5545233 9- 10077.696
\ , ',f 3. 12812904 6. 2000376 10, 36.926037
^/u ' •' Taking the first of the preceding numbers, and selecting the first root figure, 3, we take its cube
44361864 I 354. I from the period, and to the remainder attach the next period. Find the second
/^ . 27 I Ans. root figure by dividing the partial dividend, save the two right-hand figures, by
triple the square of the first root figure. Choose the last figure according to the
17.361 I character of the terminal figure of the power.
27
75
In the same manner, the cube root of
134)217728 is 1 512,
125 I Ans
92,17
So, the. cube root of
For a partial divisor we take three times the square of the first root figure,
and again choose the last figure.
1 28 1 2004 is I 234, found
thus: Partial divisor 12 48,12 The last figure is simply chosen.
The cube root of 12:812904 is 2,34, the same in form but different in value.
MATHEMATICAL ROOTS UPROOTED.
{ I
3 I 45-45 Dividing and allowing for a completed divisor, we get 7 for the second root figure,
and choose the last, which must be 7. Why?
The ^2000376= 126. Ans. / -^> •- f '
Partial divisor 3
!4-
10.00 Why is the last root figure 6?
No other figuring than the above is necessary.
Divisor 52x3=75
The ^153990656=536.
125
Always triple the square of first root figure for ^pktt&l divisor of all
289.90
but the two right-hand figures of the partial dividend.
The ^387420489 = 729.
I 343
Divisor 147 | 444.20 Having obtained the second root figure, we choose the last unerringly.
Why is it 9 ? How is 147 obtained ?
21.6.
The ^10077.696:
I 8
Divisor 12 | 20.77 O"^ the same principles the cube root of
36.926037
27
99.26
3-33.
and is found thus : 27
On the law of the terminal figure of the power depends the secret of this new method with cube
root. It is worth much to the student. Let him verify all of the above answers, by going through the
actual work in every case, and thus acquire the needed familiarity.
THIRD POWERS.
K
^^K Involving periods of noughts at the left or right of the significant figures, and extracted by the
^^B. foregoing rule.
i
The ^.000520476129 = .0809.
512
Partial divisor 192 174-75 The partial divisor is not contained in the brought-down
dividend shorn of its two right-hand figures, and we place a nought for the third figure of the root, and
arbitrarily choose the last figure which is 9.
The ^.048228544000 = .3640.
27
Divisor
27
212.28
Dividing, we find the partial divisor is contained in 212 only 6
^ times, allowing for the effect of a finished divisor. ' We choose the remainder of the root figures.
.0192.
The ^.000,007,077,
I
Divisor 3 60.77 Dividing by the partial divisor, we find it will go into the par-
tial dividend the largest possible number of times. (No divisor can go more than 9 times.) The last
figure of the root is 2. Why ?
The ^.000618470208:
512
Divisor
192
.0852. Ans.
1064.70
I. How is the partial divisor, 192, obtained?
We select, unerringly, the last root figure.
MATHEMATICAL ROOTS UPROOTED.
ADDITIONAL EXAMPLES FOR THE STUDENT.
Find the cube root of the following, viz.:
Find the value, also, of these :
I. 84604519.
7. f 13481272
2. 2803221. ^
8. 1^8615.125000
3- 3176523.
9- f 738.763264
4. 382657176.
10. f 561.515625
5. 40.353607.
II. -^ 21024576
6. 1520875.
12. if 67917312
In solving these examples, let it be understood that our only rule for cubes of three periods is, to
take out by inspection the root figure of the first period, and, having taken its cube from that period
and attached the next peiiod to the remainder for a new dividend, to find the next root figure by divid-
ing the partial dividend by triple the square of the first root figure, and arbitrarily ckoose the last figure
of the root, according to principles already explained.
Perfect Cubes of More Than Three Periods.
We will now extract the cube root of some numbers of more than three periods, and show that the
new method applies to them, with a very slight amount of additional work.
Let it be required to extract the cube root of the following numbers, viz.: 8024024008, 10460353203,
98867482624, 1 226 1 5327232, 1 54480441 6. Taking the first of the above numbers, we proceed as with
powers of three periods, thus :
8,024,024,008 I 2002. Ans,
Divisor 12.00 24.024 Explajtation.—Ymdimg the first root figure, deducting its cube, and at-
taching the next period for a dividend, we find that the partial divisor, 12, is not contained in the par-
tial dividend, 24, of the second period, and we attach the next period. The root thus far found is 20,
and triple its square is 1200, the next partial divisor, which is again not contained in the brought-down
dividend shorn of the two right-hand figures, and the root now found is 200, and we choose the last
figure.
The ■
12.6. 1
10460353203
8
24.60
1261
2187. Ans.
212X3=1323 11993.53 Explanation. — We divide, as usual, the first partial dividend by
the triple squai-e of the first root figure, which is 12, and obtain i for the second root figure. We finish
the divisor, 12, by adding to it, successively, advanced one place to the right for each addition, the
triple product of the two root figures, and the square of the last One. This finished divisor we multiply
by the last root figure, and take the product from the brought-down dividend. We need only a second
partial divisor, the triple square of 21, to find the third root figure, and we choose the last figure of
the root, according to the character of the terminal figure of the power.
4624.
The ^98867482624 =
64
Ans.
Divisor
48.36
72
348.67
33336
Fin. div. 5556 15314.82 Using the same finished divisor as an approximate divisor, we find
the next root figure to be 2, and then we select the last. Observe that 36 is put in the niche of the other
two parts of the divisor, to save space.
Obs. — To insure accuracy in finding the third root figure, it is generally best to take for a divisor the
triple square of the first two root figures. Even then there is an immense saving of work, time and
space over the old methods. By the modes of solution practiced heretofore, as treated in the books, the
work of the above example is absolutely overwhelming, covering, with the rule and the explanation, from
three to five pages. Let the pupil, for the present, accept on trust such parts or features of the rule as
he may not thoroughly understand. Further elucidation will be given in due time.
MATHEMATICAL ROOTS UPROOTED.
1st Divisor
48.81
^122615327232 = 4968.
64
586.15
53649
5961
49663.27
43218
2nd Divisor 49''X 3 = 7203 . , ,. .
6445 It is only necessary to frame a second partial divisor
to obtain the third root figure, and then we arbitrarily choose the last figure of the root. What have
we saved by this abbreviated method ?
We have saved the completion of the second divisor, the formation of the third, the completion of
the third divisor, and all the multiplications and subtractions connected with these last periods, the
prolixity and difficulty of which rapidly increase, under old methods, as we approach the end.
The 1^1544804416 = 1156. Ans.
1st divisor 3 I
Finished divisor 331
2d divisor 363
544
331
2138.04
.1815 There is no necessity for multiplying the second partial divisor, 363,
by the third root figure, 5. We do so here, to show that the remainder of the dividend divided, is less
than the divisor.
The ^12,521,107,822,861 =23221. Ans.
1st divisor 12
Fin. divisor
2nd divisor
1389
45.21
4167
Explanation. — We complete the first partial divisor, and take
its product, with the corresponding root figure, from the l^rought-
I3174 down dividend. The triple of the square of the root now found
is a partial divisor by which all the other root figures may be
367 found, except the last, and that is simply chosen. Having framed
317 the partial divisor, 1587, we ascertain how often it is contained in
the corresponding partial dividend (always excepting the two
50 right-hand figures), and multiply the divisor by the quotient fig-
ire, and subtract the result from the portion of the dividend divided, as in ordinary division. Having
ound the third figure of the root, we use for a dividend the remainder of the last partial dividend, and
lor a divisor we use the previous divisor shorn of its right-hand figure. And, in multiplying this divisor
by the quotient figure, we reckon in the number of units that would be to carry from the figure cut off.
If there were more periods than five, the process might be continued, by dropping figures, successively,
from the right of the divisor. But the last figure of the root is always chosen. And, now, in what is
said above, respecting the finding of some of the root figures by ordinary divison, lies the germ of the
method herein treated, for the approximate extraction of the cube root of surds, to be explained in
due time.
Take one more example involving five periods. Let it be required to extract the cube root of the
number
599)183,710,672,625 1 84305. Ans.
512
1st divisor 192.16
20176
Bsor 842x3 = 2116.8
I
oot fieure. which, fror
871.83
80704
Elucidation.— E-aixTiCt in the usual way to find the
first two figures of the root. Then, as will be seen by in-
64797.10 specting the work, there is a necessity for finishing only
63504 one partial divisor, and afterward forming another one
from the first two root figures. This second partial divisor
1293 we use to find two more root figures, and then choose the
figure, which, from the character of the terminal figure of the power, must be 5.
Ods.—ln the above solution, there is an immense saving of labor, time, and space. The student
cannot reahze the wide difference, if he has not gone through the labyrinths of the old methods, and
the mazes of the old rule, in solving such problems. The ru?e and the explanation, under the old sys-
tem would cover several pages of this work. By the new method here taught, we save the completion
ot the second partial divisor, and the formation and completion of two other divisors, which it is the
most desirable to obviate, because they become very large toward the end of the extraction, requiring
much time and care in the work. But, for the encouragement of the student, it may be stated that few
^authors give numbers involving more than four periods; and it is also a rare thing in applied mathe-
MATHEMATICAL ROOTS UPROOTED.
matics to find questions involving the roots beyond the fourth or fifth decimal, which are exceedingly
easy by the method here taught, but long and tedious by the old method.
ADDITIONAL EXAMPLES FOR THE STUDENT.
Find the cube root of these numbers, viz.:
1. 2176782336.
2. 87824421 125.
3. 43132764843.
Solve, also, the following :
7. ^£TO5
8.
9-
132963364864.
225199.600704.
754863.574332608.
f 17327 4H
f2^%
Note. — All perfect cubes of only two periods are to be solved at sight
reduced to improper fractions, or to mixed decimals.
Find the value of this express ion, viz.;
f T6"6 -- f 6i- (4 X f Tsl^) =
Solution: 162 -f 4 — (4 X .8) =
256 -r 4 — ( 3-2 ) =
64 — ( 3.2 ) = 60.8. Atu.
Find the value of these, viz,:
Mixed numbers should be
f54.872
^ 21.952
^ 2326. 203125
^64.964808
3- f2357947T^ff
Answer to last : J^f . Let the student find it.
Note. — If the terms of a fraction are not perfect powers of the indicated root, let them be reduced
to such powers, where possible, before the extraction begins.
The foregoing examples must suffice for illustration of the best method of extracting the root of
exact cubes. The roots of higher powers will be discussed in connection with logarithms.
CUBIC SURDS.
We will now present a brief, easy, and practical method of treating imperfect powers of the third
degree. In advance, we state that that method is, of course, one of approximation. Here it is impos-
sible to choose the final figure, since there is no definite final root figure in surds.
Let It be required to find the cube root, correct to four decimals, of the fraction
%. = .500000000 I .7937+
ist divisor
Finished divisor
2d divisor
147.81
189
1 667 1
1872.3
343
1570.00
150039
69610.00 This answer is true to the fourth place, inclusive,
56169 as verified by logarithms. We extract in the usual
way until we obtain half the number of root figures
13441 desired. We then form the second partial divisor,
13 106 and with it obtain the other two root figures by a
mode of contracted division, thus: Having found
335 the third figure of the root and taken its product with
the divisor from the partial dividend, use, instead of attaching other periods, the same partial dividend,
and drop one figure from the right of the divisor last found, and ascertain how often it will go into the
remainder of dividend accruing from the previous division, reckoning in the number that would be to
carry from the figure dropped.
MATHEMATICAL ROOTS UPROOTED.
1st divisor
Find the ^.27 correct to four places.
108.16
72,
"16
I22&.»
.270000000 I .6463-f-
216
540.00
2d divisor I228t* ' Observe that 16 is written in the niche of the
other two parts of the divisor, to save space.
Explanation. — 108 is the partial divisor. (How
obtained?) This divisor gives the next root figure.
1 1 536 is the finished divisor. (Hov/ obtained ?) Ob-
serve that the two added parts, 72 and 16, are each
advanced one place to the right. 12288 is the second
partial, or approximate, divisor.
1. How is it obtained ?
2. In the product of the root figure, 3, with the approximate divisor, 1228, account for the figure
6 in the result.
Obs. — Outside of mathematical astronomy, where, in a few instances, great precision is required, it
is scarcely necessary to approximate a root beyond four decimal figures.
The ^.640000000 &c. = .8617+
7^6q.oo
73728
4832
3686
1 146
1st divisor
192.36
144
20676
512
1280.00
124056
2d divisor 2218.8
Observe that 36 is written in the niche of the
other two parts of the divisor, to save space. 20676
is the only finished divisor it is necessary to make.
1720 After finding the third root figure, we add no other
periods to the partial dividend, but use the same dividend, and use, as a divisor, the previous one shorn
of its final figure. If we wish to find other root figures, we drop other figures from the divisor, one by
one, and continiie to divide as in common division. But it is well to bear in mind that if v/e wish to
obtain accurately a definite number of decimals in the root, we must extract in the ordinary way, until
we have obtained one-half of them, and then make a partial divisor from the root thus far found, and
use that as an approximate divisor, to find the other root figures. But this is a very great saving, as it
is the final periods that are to be dreaded in extraction. Do not fail to be familiar with the cubes i, 8,
27, 64, 125, 216, 343 , 512, 729. Otherwise you cannot select at sight the first root figure.
~- ' ~" -8735+
1st divisor
Thef^/^ =
192.49
168
divisor
20929
2270.7
.666,666,666, &c.
512
1546.66
146503
81636.66
68i2i
135 1 5 See that 49 is written in the niche again, i'nstead
1 1353 of being written thus: 192
168 which would occupy
2162 49. too much space.
Obs. — In completing any divisor, we must advance each part one place to the right.
The f .097672831790877 = .46053+
1st divisor 48.36 I 64
»72 —
I 336.72
5556 33336
2d divisor 63480.0
3368.31 7.90
3174000
I 943 I 7
190440
3877
This is accurate as far as extracted ; and, ye ,
there is a necessity to frame only a second approxi-
mate divisor. Referring to the third root figure, we
see that the divisor is not contained in the partial divi-
10
MATHEMATICAL ROOTS UPROOTED.
dend shorn of the last two figures, and we put a nought in the quotient, and two noughts to the right
of the divisor, because the squaring of the root thus far found, namely 460, would give two noughts in
the resulting partial divisor, 634800. Attaching another period, we use this divisor to find the remain-
ing root figures.
Required the indicated roots of the following surds, accurate to four deci mals, vi z.;
I.
f.171467
3-
^2.42999
1^19-44
4-
f^.571428
5-
6.
7-
f 5
f 4>^
f 42f
8.
9-
10.
f 22.4
f 3
1^41-502
f II
1^9^
f 9
f 7
f48x 4=^-182
^'300484
^129.009
^.6748
f 6
.2482
These examples must suffice for illustration of the brevity secured by this mode of approximating
the cube root of surds, by which at least three-fourths of the work necessary under other methods is
saved ; while, in perfect cubes, nine-tenths of the usual work is obviated.
HIGHER ROOTS.
To "roots of all powers," so called, but a small space can be allotted in this brief work. Some
authors, with what seems to be a strange love of novelty, rather than a desire for utility, have made
quite an array of numbers requiring the 5th, 6th, 7th, 8th, 9th, lOth, I2th, 15th, l8th, 20th; and the
25th root, to be extracted. Now, it is needless to say that no such roots occur in nature, or in the
course of applied mathematics. It is rare, indeed, in applied mathematics, that a number or quantity
occurs requiring a root beyond the ihh-d or fourth. Then why should such numbers encumber that
most practical of all the mathematical branches — arithmetic ? One of the many authors on arithmet-
ical science, whose works are in extensive use in this country, requires the 20th root of 617, the 15th
root of 15, and the 25th root of 100. Wherefore? we ask ; what the need ? When will the necessity
for their use arise? Such novelties are incubuses on the science of numbers, and ought to be relegated
to the closets of defunct mathematics. But, if mathematicians must put such impractical problems in
their books and have them solved, let them be solved by shorter and better methods than those pre-
sented in their works. That briefer and better method is by m.eans of logarithms. Especially is this
true for roots whose indices are not factorable into the square and cube roots. Indeed, even in this
case, the logarithmic method would be far preferable, and, if once adopted, would supersede all other
methods for the higher roots. For, although the 8th root can be taken by three successive extractions
of the square root, the 9th root by two successive extractions of the cube root, and the 6th root by the
cube root of the square root, or the square root of the cube root, still these roots can be much more
easily and quickly taken by logarithms. We simply take, from a table of logarithms, the log. of the
number whose root is sought ; divide this log. by the index of the root, and find, in the same table, the
number corresponding to l^^^uotient, and it will be the required root.
Ex. — Find the cttb^oot of 1.577635 — .
The log. of this number is .19800+ ; divide it by 9, the index, and the result is .02200-{-, and the
number in the table corresponding to these figures is 1. 05 19, the required root. The same result is also
easily obtained by the abbreviated method for cube root, thus:
1st root. 2d root.
1-577635= 1.1641+ I I.05-I-. Ans.
Divisor 3.3.1 i i
Divisor 363.36
iq8
Divisor
383^6^
403. 6. J
5-77
331
2466.35
229896
16739
16147
3.00
1. 641. 00 This is the answer to the example in the
1500 , work from which it has been drawn ; but it
will be seen that the logarithmic method
141 gives the answer more accurately. We have
simply taken the cube root of the cube root
by our method.
592
404
MATHEMATICAL ROOTS UPROOTED.
Find the 7th root of 308. The log. of 308, page 6 of the logarithmic table, is 2.48855 ; divide by
the index of the root, 2.48855 4- 7 = .35550-]- ; and the number corresponding to this logarithmic quo-
tient is 2.26729 + , the 7th root of 308. All that is necessary in order to extract, by logarithms, atty
root, is to look in a table of logarithms for the log. of the number to be extracted ; divide this by the
index of the root, and find, in the same table, the number corresponding to the quotient, and it will l)e
the root sought. This method is vastly shorter, and only requires a little knowledge of logarithms, and
a little facility in their use, to enable one to evolve, with despatch, all roots. But such roots belong
rather to higher mathematics. We would advise that, by all means, all roots above the third be taken
by means of logarithms. It is but an hour's work to teach, to anyone that can multiply and divide, the
use of the table. The after work is simply routine, and much valuable time is saved.
As a matter of curiosity, we give below the 7th root of the same number, as presented by its author
in one of the books of the day. That is, the 7th root of 308.
OPERATION,
f 338"= 2.59 +
■1/308=2.04-1-
2.59-1-2.04 = 4.63
4.63 -=- 2 = 2.31 -|- assumed root.
2.316 = 151.93
308 -M51. 93 = 2-0272 -h
2.31X6-1-2.0272=15.8872
15.8872 -f 7 = 2.2696 1st approximation.
2.2696® = 136.6748
308-^136.6748 = 2.253452-}-
2 2606 V 6 -^ 2 2^'i±<2 —It; 87io-;2 ^'^ ^^^'^ reached the second approximation, Con-
2.2090 X -h 2.253452 -15.671052 g^j.^^^ thought ! And these are only indicated results,
15.871052^7 = 2.267293 2d approx. none of the multiplications, divisions, etc., being car-
ried out. Now, if this belongs anywhere, and there is doubt of its having a place in applied mathe-
matics, it belongs to higher mathematics. Such skirmishing in figures is calculated to keep one hum-
ble, by giving him a modest estimate of his attainments in evolution.
EVOLUTION BY LOGARITHMS.
Required the 25th root of 100. The log. of 100 = 2.000000. 2.000000 -f- 25 = .080000, and the
number corresponding is 1.202266 + , which is the root sought. The solution of the same example, as
given by a standard author, is as follows :
Now, as the The y/ 100 =10
25th root must / 100 = y 1 =3.1 622
be less than the y 1 00 = / 3.i622^ 1.7782
24th root, let us |^'ioo= ^1.7782=1.2115
take 1.2 = the assumed root.
1.224 = 79.49684-]-
100 -^ 79,49684= 1.25792 -f
1.2 X 24 -f 1.25792 = 30.05792
30.05792 -T- 25 = 1. 2023168 1st approx.
1.2023x682* =83.2677184
100-^83.2677184= 1.2009492-f-
1.2023168 X 24-j- 1,2009492^=30.0565524
30.0565524 -^ 25 =1.202262-]- 2d approx.
We breathe a sigh of relief. Of course, not much space can be devoted, ii^ this small work, to such
solutions. It is only to show the difficulty of the subject under the old methods, in contrast with the
brevity and facility of the new, that we allow a little space for some solutions under existing methods.
Find, by the logarithmic method, the 6th root of 25632972850442049. The log. of this large num-
ber is 16.408800. Dividing by 6, we get 2.734800. The number found in the table, corresponding to
this quotient, is 543, the sixth root of the above number. The foregoing number, treated by the new
metho d, gives, for the square root, 160103007, and the
^160103007 = 543
125 Explanation. — With the approximate divisor, 75, find the second
figure of the root, 4, and choose the last. Why is it 3 ? Thus, the
75 351-03 work of the heretofore difficult cube root is almost annihilated.
12 MATHEMATICAL ROOTS UPROOTED.
What is the 20th root of 617 ? The log, of 617 is 2.790285. Dividing by 20, we have .139514,
and the number corresponding is 1,378841, the ans.
Find the 5th root of 5, The log. of 5 is .69897. Divide by 5, and get .13979, and the number cor-
responding is 1.37973, the 5th root of 5.
The 5th root of 120 is: Log. 120 = 2^piS= .41583 -(- ; and the number corresponding is 2.605174:5
the root wanted. ^ ^
Let the student find, by logarithms (see explanation of use of the table, pp. -62 and^, etc.,) the
roots of the following numbers, viz.:
1. The 8th root of 109951 1627776,
2. The I2th root of 16.3939.
3. The i8th root of 104.9617,
4, The 7th root of 1.95678.
5, The loth root of 743044.
6, The 3rd root of 4330747. '
Find, also, by logarithms, or by the abbreviated method, at your option, the cube root of the fol-
lowing numbers, viz.:
7. 7023 1 089 1 84307 2. I 9. 10964743589696.
8. 744935304423023. I 10, 1881365963625.
Required the solution of these examples : '^"^'
1. If a ball 10 inches in diameter weighs 125 lbs,, what is the diameter of a ball that weighs
216 lbs.?
Solution.— f \2i^ : ^216 :: 10 : ans= ^f|| X lO == 12,
5:6:: 10: ans, Ans. 12 inches.
2. How many balls % inch in diameter will be required to make a ball i inch in diameter ? Ans.
64 balls.
3. Suppose the diameter of the earth to be 7912 miles, and that it takes 1404928 bodies of the
size of the earth to make one as large as the sun, what is the diameter of the sun ?
^i3o|2lsx 7912= 112 X 7912 = 886144 miles. Ans.
4. A bin is 8 feet long, 4 feet wide, and 2 feet deep ; what is the linear edge of a cubical box that
will hold the same quantity of grain ?
t^S X 4 X 2 = f8x8 = 2X2 = 4 feet. Ans.
Let the curt processes be used. Extract the factors of products in preference to taking the roots
of the products.
5. If a stack of hay 24 feet high weighs 27 tons, what is the hight of a stack weighing 8 tons?
if Ty X 24 = 2^ X 24= 16 feet. Ans.
6. If a bell 4 inches high, 3 inches in diameter, and "% of an inch thick, weighs i pound, what are
the dimensions of a similar bell weighing 27 pounds ?
Ans. 12 inches high, 9 inches diameter, and ^ of an inch thick.
7. If a loaf of sugar 10 inches high weighs 8 pounds, what is the hight of a similar loaf weighing
I pound ?
fyiY, 10 = j^ X 10= 5 inches. Ans.
8. There is a bin 32 ft. long, 16 ft. wide, and 8 ft. deep; what must the side of a cubic bin be
that shall contain the same quantity ?
^32 X 16X8= 1^64X64 = 4X4= 16 ft. Ans.
9. What must be the side of a cubic bin that shall hold 350 bushels of grain ?
SOLUTION.
2150,4X350 = 752640 I 90.96+ =7 ft. 6,96 in.
Divisor 24300.81 729 Ans.
2430
236,400.00
22089429
Fin. div. 245438.
1550571 We use the finished divisor, shorn of the final fig-
, . 1472629 ure, as an approximate divisor to obtain the fourth
figure of the root. For practical purposes, the above
77942 is a close approximation.
10. If a sphere of gold i inch in diameter is worth $100, what is the diameter of a sphere that is
worth $6400 ?
iff^Q = ^64 X I = 4 inches. Ans.
11. The cubic metre is 61026.048 cubic inches ; what is the linear metre ?
Ans. 39.37 inches. Find it by approximation.
We give one more illustration, each, of the new method of taking the cube root of perfect and im-
perfect third powers :
MATHEMATICAL ROOTS UPROOTED.
1st divisor
2d divisor
The ^146113369163 = 5267. Ans.
75
30-4
7804
125
211.13
15608
55053-69 There is no necessity of the last multiplication, 6 times
48672 the second divisor. Satisfy yourself that the remainder will be
less than the divisor, and then choose the last root figure, thus
6381 saving a vast amount of work. Why is the final root figure 7?
What is the value of 1.05I to 6 decimals?
The log of 1.05 is .021189. Divide this by 3, and multiply the result by 5, and we get .035315 ; the
number corresponding is i. 084715 + , the answer. To solve the above example in the old way, will re-
quire about 30 minutes; by the logarithmic method, 2 or 3 minutes.
The ^1.1810108914205625, by the approximate method, accurate to 6 decimals, is 1.0570234;.
Find it. ^ ^^ , p ^ ^ ,-^^ T v^ A i V 5 ; .:, ; ]. OSI^O ^ :i , ^> / s^ % -■ ^ > ^ :"■
We have presented an unusually large number of solutions, in order that the cube root method
herein set forth may be clearly apprehended by all. For, knowing its advantage in brevity and sim-
plicity, and, consequently, its economy of time and space, we are thoroughly convinced that, if once
adopted, it will be abandoned for no other. Let it be remembered, that if, in approximating the cube
root of a number, it is desirable to extend the answer to a given number of decimal figures, one-half of
all the root figures must be obtained by extraction in the usual way. The other half may be obtained
by contracted division. For instance, in example 9 of the 12th page, 90.9 is obtained by extraction,
and 625 is obtained by contracted division. If it be asked how we determine when a number is a per-
fect cube, and when it is a cubic surd, we answer that, in actual business, in applied mathematics, this
fact is always known when the problem occurs in the course of our work. Perfect cubes are in the
minority in the course of mathematics. The small number of problems given under the head of evolu-
tion by logarithms, will be sufficient to illustrate the subject. The student may work any, or all, of the
others by logarithms, if he chooses. For this purpose, a table of logarithms is appended, calculated as
accurately as possible to five decimal places. The table is extensive enough to enable us to find the
roots of all numbers correct to five decimals. An explanation of the use of the table is also appended.
Should anyone desire further aid in the matter of a knowledge and ready application of logarithms in
the extraction of roots, the author will take pleasure in rendering all the assistance in his power. In
conclusion, if any have their pet theories, methods, or processes, in cube root, to which they cleave
with such a blind adherence that they cannot, or will not, see merit in anything else, this book is not
made for them. If any are moved by prejudice,or jealousy, or envy at my good fortune, or by a spirit
of criticism, or are unduly inflated with the importance of their own knowledge of the subject, through
the belief that nothing new can be presented in evolution, the book is not written for any of these classes.
All true science consists, not in the discovery of any new truth, but in the right application of ex-
isting truth, so as to render it subservient, in the highest degree, to the interests and to the pleasure of
mankind. If *' brevity is the soul of wit," it is no less the key to successful business. The large cur-
tailment of the amount of work done in book-keeping in the last few years, is only in harmony with the
spirit of the age, and, reinforces the sentiment of "short profits and quick sales." So must our
methods and processes in education be constantly improved and refined, so as to be the most highly
contributory to the important interests of business and of society.
14 MATHEMATICAL ROOTS UPROOTED.
Simple Interest.
Owing td the universal application and great practicality of this subject, we have thought proper' to
give it a place in this work, and a treatment that, for brevity, utility, and simplicity, is in keeping with
the constant drafts made upon it by all classes of men — those of inferior, as well as these of superior,
attainments. The subject is what the name signifies, but is made rather complex by some authors and
teachers, owing to the multiplicity of rules and tedious methods of treatment used by them. In simple
interest, there is scarcely a necessity for more than one uniform rule, whatever be the rate or the time.
Let us take some examples, by way of illustration :
What is the interest of $450.87 for i yr. 7 m. and 9 da., at 6 per cent.?
Operation. — 450.87
.0965
.005 225435
450.87 19.3 >&6. 270522
X — X — = 405783
I ^•i, I 9 days is .3 of a month, and the process
Ans. 43.508,955 is simply one of cancellation.
Find the interest of $125.16 for i yr. li m. 25 da., at 6 per cent.?
Process. — 1.4298
10.43
10.43
'■^ 23.83+.06 42894
-X — X — == 57192
I -13. I 14298
One year and Ii months are 23 months, and
An%. 14.91 25 d. are .83+ of a mo. So that the time is
23-834- mos., or I2ths of years, at the given rate per year. Let the cancellations, and all the multi-
plications possible, be done mentally.
Find the interest of $1500.60 for 2 yr. 4 mo., at 6^ per cent.?
125.05
1-75
125.05
>5aQ.6o 28 .06 >( 62525
X-X = 87535
I ^ta., I 12505
We take 6 times 28, plus the % of 28,
Ans. $218.8375 mentally, which gives 1.75.
Find the amount of $3050 for 4 yr. 8 m., at 51^ per cent.?
i.245>^
3050
14 .0175
3050 ^ ^e5^ 62250
X-X = 3735
I -« I
-3- Ans. $3797.250 In making multiplications mentally, after
the cancellations have been made, let the smallest numbers be so multiplied. Thus 14 times .0175, ^"^
then add i to the result, to get the amount of $1 for the lime, at the given rate. This result is then
multiplied by the principal.
MATHEMATICAL ROOTS UPROOTED.
15
Required the interest of $250 for i yr. 10 m. 15 da., 6 per cent.
125 .01 .225
--^^ 22.5 rD6- 125
I "T^ I II25
•'^ 2700
Ans. $28,125
If the amount had been required,
we
should have
proceeded thus :
.005
1. 1125
250 22.5 ^36
250
I -t^ I
556250
4ns,
22250
$278.1250
After the mental multiplication of the
time and rate, add one to the result, be-
fore the final multiplication by the prin-
ciple.
Find the interest of $51.10 for 10 m.'and 3 da., 4 per cent.
.01 51.10
51.10 lo.i Tc^. .101
I r^ I 511
3 5"
3|5-i6ii
Ans, 1.72
It may also be done thus :
=====
17-033+ -01
17.033
-3tTT& 10. 1 Aft^
.101
X X — =
I ^s^ I
17033
^
17033
As before $1.720333
What is the interest of $175.40 for 15 m. 8 da., 10 per cent.?
, 1.272+ 175.40
175.40 >5,.i66+ .10 .1272
X X — =
I ^Ki I 3508
12278
21048
Ans. $22.3108
The multiplications are made mentally,
except one.
Required the amount of $1500 for 6 m. 24 da., JJ^ per cent.
7 -025
1.0425 =amt. of $1.
1500
1500 ~'&r5 >f5,
I T-3 I
:3
$1563.7500 Ans.
Required the amount of
1.25 for I yr. 5 m. 10 da., 6}{ per cent.
8664
361
1.444+
84.25 >>^+ .o6X_
I >2. I
1.09025
84.25
545125
218050
436100
872200
Ans. . $91.85
l6 MATHEMATICAL ROOTS UPROOTED.
After first multiplication, add I to the result, before the formal multiplication.
Find the interest of $112.50 for 3 m. I da. <)% per cent.?
.2527+ 2274 112.50
112.50 ^TS954- .09^ 126 .02400
I 1^ I 4500
2250
Ans. $2.70
In multiplying the time and rate together, allow for the number of units that would be carried
from 9 times 7.
What is the interest of $408 for 20 da., 6 per cent.?
.333 .01 408
408 ":64i44- 7o§-. .00333
1224
1224
1224
1. 35864=$!. 36, Ans.
Always divide the days by 3, since every three days is tV of a month, thus reducing the days to
decimals of a month. More than 5 mills should be called a cent. Less than 5 mills should be disregarded.
Required the interest of $50 for i yr., 3 m., 27 da., 3 >^ per cent.
25 5-3 .05 1-25
-5€- -f^rf- .-K-^ 5.3
I -t5- -3- 375
-6- 625
^ ^ 3 16^6^
Ans. $2.2o8>^
Find the interest of $384.50 for 2 yr., 8 m., 4 da., 8 per cent.
10.71 .02 384.50
384.50 -:5^~iad- -"^^ -2142
I -«- I 7690
3- 15380
3845
7690
5$82.35990=$82.36. Ans.
Solve this by other methods and see if you get a different answer.
Required the amount of $275 in 4 m., 25 da., 7 per cent.
.4027+ 1. 02818
275 >>S5i+- -07 275
— X X — =
I -tt- I 514090
719726
205636
$282.7495o=-$282.75. Ans. ^^ ^ ^ , .v 1 *
After multiplying .4027 by .07, we add i to the result, for the amount of $1, before we make the last
multiplication. Multiply .402 by .07, but carry from 7 times 7.
Interest of $318.29 for 9 m., 10 da., "]}( per cent.?
•777+ 5443 318.29
318.29 -9:395;^ .07X 194 .05637
X X = —
I -in- I 222803
95487
190974
159145
Ans. 17.94.20
MATHEMATICAL ROOTS UPROOTED.
17
Required the interest of $4684.68 for 11 da., 12;^ per cent.
390-39
.04582
390.39 .183+
,+§S:;:cS. -rS^-l- .25 — -—
X X = 78078
I -K- -2- 3I23I2
I95I95
I56I56
$17.8876 = $17.89. Arts.
In making the mental multiplication by
25, allow for the number that would be to
carry, had the decimal .183+ been extended
one figure further.
Find the interest of $127.36 for I yr. 6 m. 21 da., 4^ per cent.
5-306+
5.306+ . 1.683^
-is?.^^- 18.7 .09 15918
X X = 42448
I -r^ -2- 31836
5306
Ans. $8.93. $8.929998
Find the amount of $723.60 for 2 yr. 3 m. 18 da., 5^ per cent,
2-3
723.60 ^Jfr^. .23 .529
X — X— = — = .13225
I -f~ 4 4
Add I
1. 13225
723.6
679350
339675
22645b
792575 After cancelling and multiplying the expres-
sions of time and rate, we add i to the product,
$819,296 Ans. to get the amt. of $1. This saves time and one
operation in the work. Now, if the work is short with these peculiar and mixed rates, it is much mo-e
so with all ordinary rates. We will take only two or three illustrations :
Find the interest of $780.26 for 90 da., without grace, at 1%. per cent, per month.
Operation, — 195.065
195.065 .15
;So.x6 — 3^ .15
— X— X— == 975325
I -rt- I 195065
_4_
Ans. $29.26975
What is the interest of $845 for i yn 10 m. 6 da., at I per cent a month?
845
.01 .222
845 2^.2 7t±-
X X = 1690
I -1-3- I 1690
1690
Multiplying by .01 simply throws the point
Ans. $187,590 two placesfurther to the left on the multiplicand.
Find the interest of 1040 for i yr. 9 m. 9 da. at 8 per cent.
.142
7.1 .02 1040
1040 -2*t5 "naS^
X X- 5680
I nr I 142
-3-
Ans. $147,680 This method is equally expeditious in reck-
oning up notes whereon partial payments have been made. Indeed, there is no department of interest
i8
MATHEMATICAL ROOTS UPROOTED.
where it may not be used with greater facility, and with much less work, than any other process. We
have chosen to call it the Cancellation Method. The advantage lies in its simplicity, brevity, and uni-
formity of treatment, there being but one process for all problems, whatever the rate, time or other
conditions. And surely this, of itself, is a great saving, to both teacher and pupil, of much labor and
taxing of memory, under the numerous methods and rules of interest laid down in the books. The
plan here presented is strictly mathematical, depending on the principle, that the annual rate on a dol-
lar, multiplied by the number of dollars in the principal, is equal to the interest. The time is reduced
to months and decimals of a month; and, then, the expression for months is divided by 12, thus ex-
pressing the time in years. Each example given in the foregoing pages, is simply a grouping of the
sum at interest, the years, and the rate. Cancellation naturally follows. We might have reduced the
time to days, dividing the number ot days by 360, thus making it express years. For instance, 2 yr. 4
m. 20 da. is 720 da. -f- 120 da. -|-20 da. = 860 da., or ||g years. But this is considerably longer, requir-
ing more work every way. The briefer the method of reckoning interest, the less liability to mistakes.
The one herein set forth,' takes the happy mean in all particulars.
EXAMPLES FOR THE STUDENT.
2.
3-
4-
5-
6.
7-
8.
9-
10.
Let it be required to find the interest on the following, viz.:
$300 for 2 yr. 7 m. 24 da., at 6 per cent.
$700 for I yr. 9 m. 12 da., at 6 per cent.
$400 for 2 yr. 6 m. 15 da., at 6 per cent.
$350 for 3 yr. 8 m. 24 da., at 6 per cent.
$450.87 for I yr. 7 m. 9 da., at 7 per cent.
$375.50 for 2 yr. i m. 8 da., at 7 per cent.
$125.16 for I yr. ii m, 25 da., at 7 per cent.
$658.25 for I yr. 2 m. 13 da., at 7 per cent. 1
$187.44 for I yr. 10 m. 24 da., at 7^,7 per cent. [-Find the amount.
$444.84 for I yr. i m. 16 da., at 5 per cent. j
Also, reckon up the following promissory notes, on which indorsements have been made, viz.:
$167.42. Selma, Apr. 15, 1882.
1. For value received, I promise to pay Judge Fowler, or order, one hundred sixty-seven and -^
dollars, in 6 months from date with interest.
Tom Scroggins.
Indorsements :
May 21, 1883, $42.18; July 17, 1884, $6.25; Sept. 9, 1884, $48.16; Jan. 27, 1885, $27.47. What
was due Apr. 15, 1886? ^ns. $72,277.
$472.76. Selma, June 4, 1884.
2. For value received of Arrents & Longacre, I promise to pay them, or their order, four hundred
seventy-two and ^^% dollars, in 6 months from date, with interest at 7 per cent, afterward.
Jno. Grubs.
Indorsements :
Apr. 10, 1885, $125,843 ; Nov. 28, 1885, $133,724; Apr. 15, 1886, $223,081. What will due Nov.
13, 1886? ylns. $24.95.
Let these be done strictly by the abbreviated process — it saves half the usual work. These exam-
ples must suffice for our book. The student will find abundant material for practice from other sources.
We will present the solution of the last promisory note, to show the plan by the cancellation process.
W^e first write the dates in succession, thus ;
1886, II, 13 =6 m. 28 da.
1886, 4, 15 = 4 m. 17 da.
1885, II, 28 = 7 m. 18 da.
4, 10 = 4 m. 6 da. JV. B. — The four cancellations and multiplications" following, present
12, 4= the entire work, and not simply the indicated vioxY'.
1885,
472.76
1.0245
236380
189104
94552
47276
$484,342
Payment — 125.843
$358-499
.63+
358.50 .i^ .07
X X
358-50
1.0443
10755
14340
14340
3585
$374,381
Payment— 133.724
$240,657
MATHEMATICAL ROOTS UPROOTED.
19
1.026635
240.657
•3805
240.657 4.566 U
X X—
I -^-^- I
07
7186445
5133175
6159810
4106540
2053270
$247.06689
Payment— ^223 .08 1
$23.99
•574-
23-99 -^^^^f -07
I -irt- I
•57 X. 07+ I = 1.0399
$24,947 = $24.95.
Ans.
ramt. of $1.
In partial payments, most authors, in illustrating their methods, give simply a brief of results, as if
they would make the work appear short. After cancellation and multiplication of the expressions lor
time and rate, let i be added to the result, for the amt. of $1, before the mechanical multiplication is
made. See above. Observe the manner of writing the dates, all at once, and all in one group, from
the latest to the earliest, and then the consecutive subtraction of them, thus giving the several periods
of time at once, before the cancellation processes are commenced. This is a great saving of time, and
promotes simplicity. And here it may be stated that, with respect to the subject of interest, the author
does not so much claim to have discovered new truth, as he does a new and right application of it.
Much of scientific truth is as good as lost, through the circuitous, obscure processes under which it is
presented.
Required the interest on the following, viz.:
1. $1284.60 for 5 m. 12 da., at ^ per cent, a month.
2. $621.09 for 7 m. 16 da., at ^ per cent, a month.
3. $818.26 for 9 m. 3 da., at \ per cent, a month.
4. $220.38 for 2 m. 21 da., at lOj^ per cent, per annum.
5. $62.96 for I yr. 8 m. 23 da., at 11 per cent per annum.
62.96
1-73, .1903
62.96 —20^76-+-
-X X =
1888
56664
6296
$11.98. Ans. to last.
$614.42.
For value received, I promise to pay D
on demand, with interest at 6^ per cent.
Indorsements :
May 15, 1886, $169.30; June 10, 1886, $88.40; Sept. I
on this note Nov. 20, 1886?
Selma, Cal., May i, 1886.
Wagner, or order, six hundred, fourteen and -^ dollars,
John Davis.
1886, $325.80. How much will be due
Write the dates in a group, as above ; begin with the date of giving the note, and subtract each
from the next succeeding, as 5 m. i da. from 5 m. 15 da., always making the subtractions mentally, and
writing the payments opposite the intervening times. Find the amount of the original principal for the
first term of time, and of each succeeding principal for its term of time, subtracting each payment, in
order, from the corresponding amount, till you come to the maturity of the note. Then find the
amount of the last principal for the corresponding term of time, and it will be the balance.
Solution of the last example : 614.42
1.00262 amt. of $1.
1886,
II,
20 = 2 m.
2 da.
PAYMENTS
1886,
9,
18 = 3 m.
8 da.
1886,
6,
10 = m.
25 da.
$325.80.
IS86,
S,
15 = om.
14 da.
88.40.
1886,
5,
I
169.30.
.0291
-.1165- .09
614.42 ~:^f6B- .2f-
X X —
I -12- -4-
122884
368652
122884
61442
$616.03
169.30
$446.73 2d prin.
20 MATHEMATICAL ROOTS UPROOTED,
1.00468 amt. of $1.
44673
20S .02^ 357384
446.73 .-853. -r^T^ 268038
X X 178692
I -«- S^ 44673
$448.82
88.40
$360.42 3d prin.
1.01836 amt. of $1.
360.42
.204 .09 203672
360.42 - 3-^6^ : >^ 407344
X X — = 611016
I -13— _4_ 305508
After multiplying .204
$367,037
325.80
by .09, prefix I to the result. $41.24 4tli prin.
I.01162 amt. of $1.
41.24
404648
202324
101 162
404648
$41,719. Ans. $41.72 Ba/.
The above is the entire work. It is comparatively short, and is quickly done. Let the student use
the stereotyped methods in use, and he will at once see a vast difference in favor of this curt cancella-
tion process, uniform for a/l rates, times, and conditions, and equally easy for all questions in interest.
For these and other reasons, this method, once adopted, will be used in preference to any other.
In reckoning up the balance due on promissory notes whereon partial payments have been made,
let the cancellations be so managed that the uncancelled factors may, as far as possible, be multiplied
iogQihev mentally ; or, at least, maybe reduced to one formal multiplication. The advantage of the
cancellation process may be seen in the following problem :
Find the amount of $235.18 for 2 yr. 8 m. and 12 da., at 5>^ per cent.
$235.18
.9 I. 144
•w>8.
235.18 :yM. -16 94072
X X = 94072
I Hri- -5-. 258698
$269.04592 = $269,046. Ans.
Observe that, after multiplying together the expressions of time and rate, .g and .16, we add I to
the result, which makes the amt. of $1. This multiplied into the principal gives the amt. of the debt.
Hence, in the cancellations, it is not proper to cut down the principal, when the amount is to be found.
If only the interest is required, factors may be stricken from any of the three parts.
Those who have not tried this method cannot realize how easily these factors (principal, time in
years, and rate) can be thrown together, and cut down to the answer, with a very small amount of fig-
uring, whatever be the nature of the parts. The years and months are reduced to months, simply by
inspection, without a mental effort ; the days are reduced to the decimal of a month, by dividing them
by 3 ; and under this mixed expression we place 12, thus expressing the whole time in years.
We have little patience with sticklers for analysis in everything. It is very essential in some depart-
ments of arithmetical science, but is utterly useless in many of its most practical subjects. As an inci-
dental feature, and as conducing to thoroughness in the fundamental principles of a mathematical edu-
cation, analysis, in a few subjects in arithmetic, should be thoroughly taught to the young, but beyond
this it is useless. In interest it is of no value. The banker, the lawyer, the real estete man, and all
other practical people, have no use for analysis in any of the several departments of interest, but must
have the most direct, curt processes for all problems arising under this broad department in the science
MATHEMATICAL ROOTS UPROOTED. 21
of numbers. Teachers should not lose sight of the fact that, in our practical civilization, we are, in
many things, reducing more and more to practicality. Hence, as business increases and ramifies into
various new departments, thus multiplying our cares and duties, we must seek to do our work with the
utmost despatch consistent with accuracy. The cancellation process in interest meets the demands of
business, on the subject to which it belongs. It supersedes the necessity and the utility of any 6 per
cent, method, or 12 per cent, method, or i per cent, method, or 10 percent method, or any other
specific method. It secures uniformity, simplicity, brevity, accuracy,
FINIS.
22 LOGARITHMS.
Logarithms.
A Logarithm is simply an exponent of a power. The logarithm of a number is the exponent of the
power to which the base of the number must be raised, to produce the number. Thus, in the following
equations :
5" = 1 and lo<* = I
51 = 5 loi = 10
52 = 25 lo* = 100
53 = 125 10* = loog,
O, I, 2, 3 are the logarithms of the respective numbers to which they stand opposite in the several equa-
tions. 5 is the base in the one set, and ten in the other. In any one of the above quotations, the value
of the second member depends on the numerical value of the base and the exponent attached. In a sys-
tem of logarithms, any number above I may be taken as a base, and, by suitably varying the exponent,
the base being unaltered, all possible numbers may be represented. For example, lo^ -82000 represents
the number 6607, and 3.82000 is the log. of this number. io3'7'*225 represents the number 5524, and
3.74225 is the log. of this niimber. It means that 10 must be raised to the power denoted by 3.74225
to produce 5524. In all practical mathematics, 10 is the base. The system is called the Common, or
Brigg's system, and, in it, all numbers, integral or fractional, are regarded as some power of 10. 10° is
no power of 10, and is equal to 10 divided by 10, or to I. That is, the log. of i is o. All numbers be-
tween I and 10 have, for their logarithm, a decimal fraction ; all numbers between 10 and 100 have, for
their logarithm, i -[-a decimal ; and all numbers between 100 and 1000 have, for their logarithm, 2 -(- a
decimal ; and so on. See, page I of the table of logarithms, in column headed N, that numbers be-
tween I and 10, 10 and loo, and 100 and 1000, respectively, fulfill the above conditions. The log. of 7,
for example, is .84510, and that of 25 is 1-39794, and these logs, are simply exponents. io''-845io __ 7^
and ioi-^9794 =25, signifying that 10 must be raised to these powers, respectively, to produce 7 and 25.
To find the logarithms of numbers over 100, and under looo. — Look opposite the number, in column
headed O, and find the logarithm. The log. of 398, page 7 of the table, is 2.59988.
.-To find the logarithms of numbers of four figures. — Look under caption iV'for the first three 'fig-
ures of the number, and at the top of the page for the fourth figure ; and opposite the one part of the
number and under the other, find the logarithm. Thus, the log. of 6982 is 3.84398. The decimal part
of any logarithm is called the mantissa, and the integral part, the characteristic. In the log. of 1840,
which is 3.26482, 3 is the characteristic, and .26482 is the mantissa. The characteristic of all numbers
between I and 10 is o.
To find the logarithms of numbers of more than four figures. — Find, for example, the logarithm of
248963. On page 4 we find, as previously directed, the «/rt«/wa*corresponding to the first four figures,
2489, to be .39602 ; and, to this partial mantissa, there must be an addition for the remaining part of
the number, 63. And since this addition affects only the decimal part, or mantissa, and not the char-
acteristic, 63, the remaining part of the number must be regarded as a decimal. This decimal, .63, we
multiply by the tabular difference, opposite the mantissa, in column D, which is 17+, or 17.5, and get
.63 X 17.5 = 11.025, giving II to be added to the final figures of the partial mantissa, .39602, already
taken out, making .39613 ; and the characteristic is 5, being always one less than the number of figures
in the integral part of the number whose logarithm is sought. Thus, the log. 248963 = 5.39613.
Required the logarithm of 142967542. The mantissa of the first four figures, 1429, page 2 of the
table, is .15503, and the tabular difference is 30+, or 30.5. This multiplied into .67542, the remainder
of the number treated as a decimal, gives 20.6, or 21, to be added to the terminal figures of the partial
mantissa already taken out, making .15524, and the characteristic is 8. Thus, the log. 142967542 =
8.15524. In making additions to the mantissa, more than 5 decimal units should be reckoned i , less
than 5 should be disregarded.
For the same figures, in the same order, the mantissa is the same, whatever the local value or the fig-
ures. Thus, the mantissa of the logs, of these numbers, viz.: 8328, 832.8, 83.28, 8.328, .8328, .08328.
,008328, etc., is .92054, the same for each of the numbers. The characteristic of the first is 3 ; of the
second, 2 ; of the third, i ; of the fourth, o; of the fifth, — i ; of the sixth, —2 ; of the seventh, — 3. The
LOGARITHMS.
23
characteristic of a decimal is always negative, and numerically one more than the number_of noughts
prefixed to the decimal. The negative sign is usually vsrritten over the characteristic, thus 2, 6 9 8 9 7, in
the log. of .05.
Find the logarithm of .6423. On page 12 we find the mantissa to be .80774, and the characteristic
is T, making T. 8 7 7 4, for the log. of .6423.
To find the number corresponding to a given logarithm. — What is the number whose log. Is
1. 681^4? Looking on page I, we find this log. opposite 48. Hence, 48 is the number whose log. is
I. 68 I 24.
Find the number having for its logarithm 2 , 3 6 3 5_. Looking on page 4, we find opposite 230 and
under 7, the number 230.7, the answer required^.
Find the number having for its logarithm 2,64367. Looking for the nearest mantissa to the given
one, we find it, page 8, opposite 440 and under 2, to be .64365. This mantissa we subtract from the
given one, and divide the difference, 2, by the tabular diff^ence, 10, and get .2. Appending this to
the 4402, already taken out, we get 44022. And now, as the characteristic is — 2, we prefix one o to
the last result, and get .044022 for the required number.
What is the number having for its logarithm .29824? The nearest mantissa is .29820, page 3, op-
posite 198 and under 7 ; ,29824— ,29820 = 4; 4-4-22, the tab. difif., gives 1.8, or 2 nearly. Appending
2 to 1987, we have 19872 — . And, since there is no characteristic, the integral part is I. Hence, .29824
= log. I. 19872.
; Find the cube root of .4986. The log. of .4986 (p. 9) is T.6 9 775, This we divide by 3, the in-
dex of the required root. But since the characteristic is negative, while the mantissa is always positive,
we cannot directly divide the logarithm by the index 3. But T,6 9 7 7 S == ^-f 2^.69775, in which the
characteristic is exactly divisible by the index 3. Dividing, we get T,?99 2 5, We now find the num-
ber corresponding to this logarithm. The nearest mantissa is .89922.' Subtracting it from the given
.89925, we get 3, which, divided by the corresponding tabular diffe^nce, §-f ' gives 5. The number
corresponding to the mantissa .89925 is 79295. To this prefix^'wic. o, to correspond to the negative
characteristic, and the cube root of .4986 is .-&7929§^ ^^ ,^ y- ^7 <^J^
When the characteristic is negative, and not divisible by me4nclex^of any root, add to it the smallest nega-
tive number that will render it divisible, and then prefix the same number, with a plus sign, to the mantissa.
What is the 5th root of 512.8? The log. of 512.8 = 2.70995 ; 2,70995 -f- 5 "= -54199 ; the nearest
mantissa, opposite 348 and 3, is. 54195; .54199 — .54195=4; and 4 -r 12^-, the tab, diff,, gives 3 to
to be apended to 3.483, making 3.4833, for the 5th root of 512.8.
k
5,if2^«02.
T
THE COMMOH OR BRIGGS LOGARITHMS
—OF THE— .
iT-A.rrxTie-A.ij 3sr"U-3iv^BEies
FROM 1 TO 10000.
I— 100.
H
loS
If
log
ir
log
N
log
N
log
I
0. 00 000
21 I
32 222
41
I. 61 278
61 I
78533
81
I. 90 849
2
0. 30 103
22 I
34 242
42
62325
62 I
79 239
82
I. 91 381
3
0. 47 712
23 I
36 173
43
63347
63 I
79 934
83
I. 91 908
4
0. 60 206 ■
24 I
38021
44
64345
64 I
80618
84
I. 92 428
5
0. 69 897
25 I
39 794
45
65 321
65 I
81 291
85
I. 92 942
6
0. 77 815
26 I
41 497
46
66 276
66 I
81 954
86
I. 93 450
7
0. 84 510
27 I
43 136
47
67 210
67 I
82 607
87
I- 93 952
8
0. 90 309
28 I
44 716
48
68 124
68 I
^3 251
88
I. 94 448
9
0. 95 424
29 I
46 240
49
69 020
69 I
S3 885
89
I- 94 939
lO
I. 00 000
30 I
47 712
50
69897
70 I
84 510
90
I. 95 424
II
I. 04 139
31 I
49 136
51
70757
71 I
05 126
91
I. 95 904
12
I. 07 918
32 I
50515
52
71 600
72 I
85 733
92
I. 96 379
13
I. II 394
33 I
51 851
53
72428
73 I
86 332
93
I. 96 848
14
I. 14 613
34 I
53 148
54
73 239
74 I
86923
94
I. 97313
15
I. 17 609
35 I
54407
55
74036
75 I
87506
95
I. 97 772
16
I. 20 412
36 I
55630
56
74819
76 I
88 081
96
I. 98 227
17
I. 23 045
37 I
56 820
57
75587
77 I
88 649
97
I. 98 677
18
I. 25 527
38 I
57978
58
76 343
78 I
89 209
98
I. 99 123
19
I. 27 875
39 I
59 106
59
77085
79 I
89763
99
I. 99 564
20
I. 30 103
40 I
60 206
60
I- 77815
80 I
90309
100
2. 00 000
N
log
N
log
H
log
H
log
V
log
N
1
%
3
4
5
6
7 8
9
1)
100
00 000
00043
po 087
00130
00 173
00 217
00 260
00 303 00 346
00 389
43
lOI
00432
00475
00 518
00 561
00 604
00 647
00 689
00 732 00 775
00 817
43
I02
00860
00903
00945
00988
01 030
01 072
01 115
01 isf 61 199
01 242
42
103
01 284
01 326
01 368
01 410
01 452
01 494
01536
01 578 01 620
01 662
42
104
01 703
01 745
01787
01 828
01 870
01 912
01953
01 995 02 036
02 078
42
105
02 119
02 160
02 202
02 243
02 284
02325
02 366
02 407 02 449
02 490
41
106
02531
02572
02 612
02653
02 694
02735
02776
02 816 02 857
02 898
41
107
02938
02979
03019
03 060
03 100
03 141
03 181
03 222 03 262
03302
40
108
03342
03383
03423
03 463
03503
03543
03583
03 623 03 663
03703
40
109
03743
03782
03 822
03 862
03902
03941
03 981
04021 04060
04 100
40
110
04139
04179
04 218
04258
04297
04336
04376
04415 04454 04493
39
III
04532
04571
04 610
04 650
04 689
04727
04766
04 805 04 844 04 883
39
112
04 922
04961
04999
05 038
05 077
05 115
05 154 05 192 05 231
05 269
39
113
05 308
05 346 05 385
05423
05 461
05500
05538 05 576 05 614
05652
38
114
05 690
05729
05767
05805
05 843
05 881
05 918
05 956 05 994
06 032
38
115
06 070
06 108
06 145
06 183
06 221
06 258
06 296
06333 06371
06408
38
116
06 446 06 483
06 521
06558
06595
06 633 06 670
06 707 06 744 06 781
37
117
06819
06856
06 893
06 930
06 967
07004
07041
07 078 07 ii5
07 151
37
118
07 188
07 225
07 262
07 298
07335
07372
07 408
07 445 07 482
07518
37
119
07555
07 591
07 628
07 664
07 700
07 737
07773
07 809 07 846
07882
36
120
07 918
07 954 07 990
08 027
08063
08 099
08135
08 171 08 207
08 243
36
121
08 279
08 314
08350
08386
08 422
08458
08493
08 529 08 56S
08 600
36
122
08 636
08 672
08 707
08743
08778
08814
08849
08 884 08 920
08955
35+
123
08 991
09 026
09 061
09 096
09132
09 167
09 202
09237 09 272
09307
35
124
09342
09377
09 412
09447
09 482
09517
09552
09 587 09 621
09 656
35
125
09 691
09 726
09 760
09795
09830
09 864
09899
09 934 09 968
10003
35
126
10037
10 072
10 106
10 140
10 175
10 209
10243
10 278 10 312
10 346
34
127
128
10 380
10 721
10 415
10755
10449
10 789
10483
10517
10857
10551
10 890
10 585
10924
i4o 619 10 653
10958 10992
10687
II 025
34
34
10823
129
II 059
1 1 093
II 126
II 160
II 193
II 227
11 261
II 294 II 327
II 361,
33+
130
II 394
II 423
II 461
1 1 494
II 528
II 561
II 594
II 628 II 661
11 694
33
131
11727
11 760
II 793
II 826
II 860
II 893
II 926
II 959 II 992
12 024
33
132
12057
12 090
12 123
12 156
12 189
12 222
12 254
12 287 12 320
12352
33
^33
12385
12 418
12450
12483
12516
12548
12 581
12 613 12 646
12 678
32+
134
12 710
12743
12775
12808
12 840
12872
12905
12937 12969
13 001
32
135
13033
13066
13098
13 130
13 162
13 194
13226
13 258' 13 290
13322
32
^3^
13354
13386
13 418
13450
13 481
13 513
13545
13577 13609
13 640
32
137
13672
13 704
13735
13767
13799
13830
13862
13893 13925
1395^
31+
^38
13988
14 019
14 051
14082
14114
14 145
14 176
14 208 14 239
14270
31
139
14301
14333
14364
14395
14426
14457
14489
14520 14 551
14582
31
uo
14613
14644
14675
14706
14737
14768
14799
14829 14860
14 891
31
141
14922
14953
14983
15 014
15 045
15076
15 106
15 137 15 168
15 198
31
142
15229
15259
15 290
15320
15 351
15 381
15 412
15 442 15 473
15503
30+
1
143
15534
15564
15594
15625
15655
15685
15 715
15 746 15 776
15 806
30
144
15836
15866
15897
15927
15957
15987
16 017
16047 16077
16 107
30
145
16 137
16 167
16 197
16227
16 256
16286
16316
16346 16376
16 406
30
146
16435
16465
16495
16524
16554
16584
16613
16 643 16 673
x6 702
30
147
16732
16 761
16 791
16820
16850
16 879
16 909
16 938 16 967
16997
29+
148
17 026
17056
17085
17 114
17 143
17 173
17 202
17 231 17 260
17289
29
149
17319
17348
17377
17406
17435
17464
17493
17522 17551
17580
29
150
17609
17638
17667
17696
17725
17754
17782
17 811 17 840
17 869
29
N
1
2
3
4
5
6
7 8
9
N
12 3 4
5 6 7 8 9
D
150
17609 17638 17667 17696 17.725
17754 17782 17811 17840 17869
29
151
17898 17926 17955 17984 18 013
X804X 18070 18 099 18 127 18x56
29
152
18 184 18213 18241 18270 18298
X8327 18355 18384 184x2 18 44X
28+
153
18469 18498 18526 18554 18583
18 6x1 18639 18667 18696 18724
28
154
18752 18780 18808 18837 18865
18893 18 921 18949 18977 19005
28
155
19033 19 061 19089 19 117 19 145
19173 19 201 19229 19257 19285
28
156
19 312 19340 19368 19396 19424
19 451 19479 19507 19535 19562
27
157
19590 19 618 19645 19673 19700
19728 19756 19783 X981X X9 838
28
158
19866 19893 19 921 19948 19976
20003 20030 20058 20085 20x12
27
159
20 140 20 167 20 194 20 222 20 249
20276 20303 20330 20358 20385
27
160
20412 20439 20466 20493 20520
20548 20575 zo 602 20629 20656
27
161
20683 20710 20737 20763 20790
208x7 20844 20871 20898 20925
27
162
20952 20978 21005 21032 21059
21 085 2X XX2 21 139 21 X65 21 I92
27
^63
2x2x9 2x245 21272 2x299 21325
21352 2x378 2x405 21431 21458
27
164
21484 21 511 21537 21564 21590
21 6x7 21 643 21 669 21 696 21 722
26
165
21 748 21 775 21 801 21 827 2X 854
21 880 21 906 21 932 21 958 2X 985
26
166
220x1 22037 22063 22089 22 115
22 141 22 167 22 194 22 220 22 246
26
167
22 272 22 298 22 324 22 350 22 376
22401 22427 22453 22479 22505
26
168
22531 22557 22583 22608 22634
22 660 22 686 22 712 22 737 22 763
26
169
22 789 22 814 22 840 22 866 22 89I
229x7 22943 22968 22994 23019
26
170
23045 23070 23096 23 121 23147
23 172 23 198 23 223 23 249 23 274
25
171
23300 23325 23350 23376 234OX
23426 23452 23477 23502 23528
25
172
23553 23578 23603 23629 23654
23679 23704 23729 23754 23779
25
173
23 8o5 23 830 23 855 23 830 23 905
23930 23955 23980 24005 24030
25
174
24055 24080 24105 2413,0 24x55
24180 24204 24229 24254 24279
25
175
24304 24329 24353 24378 24403
24428 24452 24477 24502 24527
25
176
24551 24576 2460X 24625 24650
24674 24699 24724 24748 24773
25
177
24797 24822 24846 24871 24895
24920 24944 24969 24993 25018
24+
178
25 042 25 066 25 09X 25 115 25 139
25 164 25 188 25 2X2 25 237 25 261
24
179
25285 25310 25334 25358 25382
25406 25431 25455 25479 25503
24
180
25527 25551 25575 25600 25624
25 648 25 672 25 696 25 720 25 744
24
181
25 768 25 792 25 8x6 25 840 25 864
25888 25912 25935 25959 25983
24
182
26007 26031 26055 26079 26x02
26 126 26 150 26 174 26 198 26 221
24
183
26245 26269 26293 26316 26340
26364 26387 26411 26435 26458
24
184
26482 26505 26529 26553 26576
26 600 26 623 26 647 26 670 26 694
23+
185
26717 26741 26764 26788 26811
26834 26858 26881 26905 26928
23
186
26951 26975 26998 27021 27045
27 068 27 091 27 114 27 138 27 x6i
23
187
27 184 27 207 27 231 27 254 27 277
27300 27323 27346 27370 27393
23
188
27416 27439 27462 27485 27508
27531 27554 27577 27600 27623
23
189
27646 27669 27692 27715 27738
27761 27784 27807 27830 27852
23
190
27875 27898 27 921 27944 27967
27989 28012 28035 28058 28081
23
191
28 103 28 126 28 149 28 171 28 194
282x7 28240 28262 28285 28307
23
192
28330 28353 28375 28398 28421
28443 28466 28488 28 511 28533
23
193
28556 28578 28601 28623 28646
286-68 28691 28713 28735 28758
22+
194
28 780 28 803 28 825 28 847 28 870
28892 28914 28937 28959 28981
22
195
29 003 29 026 29 048 29 070 29 092
29 115 29137 29x59 29181 29203
22
196
29226 29248 29270 29292 29314
29336 29358 29380 29403 29425
22
197
29447 29469 29491 29513 29535
29557 29579 2960X 29623 29645
22
198
29 667 29 688 29 7x0 29 732 29 754
29776 29798 29 S20 29842 29863
22
199
29885 29907 29929 29951 29973
29994 300x6 30038 30060 30081
22
200
30 103 30 12S 30 146 30 168 30 190
30 211 30233 30255 30276 30298
22
N
12 3 4
5 6 7 8 9'
N
12 3 4
5 6 7 8 9
1)
200
30 T03 30 125 30 146 30 168 30 190
30 211 30233 30255 30276 30298
22
20I
30320 30341 30^63 30384 30406
30428 30449 30471 30492 30514
22
202
30535 30557 30578 30600 30621
30643 30664 30685 30707 30728
21+
203
30750 30771 30792 30814 30835
30856 30878 30899 30920 30942
21
204
30 963 30 984 31 006 31 027 31 048
31 069 31 091 31 112 31 133 31 154
21
205
31 175 31 197 31 218 31 239 31 260
31 281 31 302 31 323 31 345 31 3^^
21
206
31 387 31 408 31 429 31 450 31 471
31492 31 513 31534 31555 31576
21
207
31 597 31 618 31 639 31 660 31 681
31 702 31 723 31 744 31 765 31 785
21
208
31 806 31 827 31 848 31 869 31 890
31 911 31931 31952 31973 31994
21
209
32015 32035 32056 32077 32098
32 1X8 32 139 32 160 32 181 32 20I
21
210
32 222 32 243 32 263 32 284 32 305
32325 32346 32366 32387 32408
21
211
32 428 32 449 32 469 32 490 32 510
32531 32552 32572 32593 32613
20+
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32 634 32 654 32 675 32 695 32 715
32736 32 756 32777 32797 32818
20
213
32 838 32 858 32 879 32 899 32 919
32 940 32 960 32 980 33 001 33 021
20
214
33 041 33 062 33 082 33 102 33 122
33 143 33 ^^3 33 ^^3 33 203 33 224
20
215
33 244 33 264 33 284 33 304 33 325
33 345 33 365 33 385 33 405 33 425
20
216
33 445 33 465 33 486 33 506 33 526
33 546 33 566 33 586 33 606 33 626
20
217
33 646 33 666 33 686 33 706 33 726
33 746 33 766 33 786 33 806 33 826
20
218
33 846 33 866 33 885 33 905 33 925
33 945 33 9^5 33 985 34 oo5 34 025
20
219
34044 34064 34084 34104 34124
34143 34163 34183 34203 34223
20
220
34242 34262 34282 34301 34321
34341 34361 34380 34400 34420
20
221
34 439 34 459 34 479 34 498 34 5^8
34 537 34557 34577 34596 34616
20
222
34635 34655 34674 34694 34713
34 733 34 753 34 772 34 792 34 811
19+
223
34830 34850 34869 34889 34908
34928 34 947 34967 34986 35005
19
224
35 025 35 044 35 064 35 083 35 102
35 122 35 141 35 160 35 180 35 199
19
225
35218 35238 35257 35276 35295
35 315 35 334 35 353 35 372 35 392
19
226
35 411 35 430 35 449 35 468 35 488
35 507 35 526 35 545 35 564 35 583
19
227
35 603 35 622 35 641 35 660 35 679
35 698 35 717 35 736 35 755 35 774
19
228
35 793 35813 35832 35851 35870
35 889 35 908 35 927 35 946 35 965
19
229
35984 36003 36021 36040 36059
36078 36097 36 116 36135 36154
19
230
36173 36192 36 211 36229 36248
36267 36286 3630^ 36324 36342
19
231
36361 36380 36399 36418 36436
36455 36474 36493 36 511 36530
19
232
36549 36568 36,586 36605 36624
36642 36661 36680 36 6g8 36717
19
233
36 T36 36 754 36 773 36 791 36 810
36829 36847 36866 36884 36903
19
234
36922 36940 36959 36977 36996
37014 37033 37051 37070 37088
18+
235
37 107 37 125 37 144 37 162 37 181
37 199 37 218 37 236 37 254 37 273
18
236
37291 37310 37328 37346 37365
37383 37401 37420 37438 37 457
18
237
37475 37 493 37 511 37 53o 37 548
37566 37585 37603 37621 37639
18
238
37 658 37 676 37 694 37 712 37 731
37749,37767 37785 37803 37822
18
239
37 840 37 858 37 876 37 894 37 912
37 931 37 949 37 967 37 985 38 003
18
240
38021 38039 38057 38075 38093
38 112 38 130 38 148 38 166 38 184
18
241
38 202 38 220 38 238 38 256 38 274
38292 38310 38328 38346 38364
18
242
38382 38399 38417 38435 38453
38471 38489 38507 38525 38543
18
243
38561 38578 38596 38614 38632
38 650 38 668 38 686 38 703 38 721
18
244
38739 38757 38775 38792 38810
38828 38846 38863 38881 38899
18
245
38917 38934 38952 38970 38987
39005 39023 39041 39058 39076
18
246
39094 39 III 39 129 39 146 39 164
39182 39199 39217 39235 39252
18
247
39270 39287 39305 39322 39340
39358 39 375 39 393 39 4io 39428
18
248
39445 39463 39480 39498 39515
39 533 39550 39568 39585 39602
17+
249
39620 39637 39655 39672 39690
39 707 39 724 39 742 39 759 39 777
17
250
39 794 39 811 39829 39846 39863
39.881 39898 39915 39 933 39950
17
N
12 3 4
5 6 7 8 9
250
251
252
253
254
255
256
257
258
259
260
26l
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
39 794
39967
40 140
40312
40483
40654
40 824
40993
41 162
41330
41 497
41 664
41 830
41 996
42 160
42325
42488
42 651
42813
42975
43 136
43 297
43 457
43616
43 775
43 933
44091
44248
44404
44560
44716
44871
45025
45 179
45332
45 484
45 637
45788
45 939
46 090
46 240
46389
46538
46687
46 835
46 982
47 129
47 276
47422
47567
47 712
398II
39985
40157
40329
40 500
40 671
40 841
41 010
41 179
41347
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41 681
41 847
42 012
42 177
42341
42504
42 667
42 830
42991
43 152
43313
43 473
43 632
43 791
43949
44 107
44 264
44420
44576
44731
44886
45 040
45 194
45 347
45 500
45652
45803
45 954
46 io5
46255
46 404
46553
46 702
46850
46997
47 144
47 290
47436
47582
47727
39 829 39 846 39 863
40 002 40 019 40 037
40 175 40 192 40 209
40346 40364 40381
40518 40535 40552
40 688 40 705 40 722
40 858 40 875 40 892
41 027 41 044 41 061
41 196 41 212 41 229
41 Z^2, 41 380 41 397
41 531 41 547 41 564
41 697 41 714 41 731
41 863 41 880 41 896
42 029 42 045 42 062
42 193 42 210 42 226
42357 42374 42390
42521 42 537 42 553
42 684 42*700 42 716
42 846 42 862 42 878
43 008 43 024 43 040
43 169 43 185 43 201
43329 43345 43361
43 489 43505 43 521
43 648 43 664 43 680
43 807 43 823 43 838
43 96S 43 981 43 996
44122 44138 44154
44279 44295 44 311
44436 44451 44467
44592 44607 44623
44 747 44762 44778
44902 44917 44932
45 056 45071 45 086
45 209 45 225 45 240
45 362 45 378 45 393
45515 45530 45 545
45 667 45 682 45 697
45 818 45 834 45 849
45 969 45 984 46 000
46 120 46 135 46 i5o
46 270 46 285 46 300
46 419 46 434 46 449
46 568 46 583 46 598
46 716 46 731 46 746
46 864 46 879 46 894
47 012 47 026 47 041
47 159 47 173 47 188
47305 47319 47 334
47451 47465 47480
47596 47 611 47 625
47 741 47 756 47 770
39881 39898 39915
40 054 40 071 40 088
40 226 40 243 40 261
40398 40415 40432
40569 40586 40603
40 739 40 756 40 773
40 909 40 926 40 943
41 078 41 095 41 III
41 246 41 263 41 280
41 414 41 430 41 447
41 581 41 597 41 614
41 747 41 764 41 780
41 913 41 929 41 946
42 078 42 095 42 III
42 243 42 259 42 275
42 406 42 423 42 439
42 570 42 586 42 602
42 732 42 749 42 765
42 894 42 911 42 927
43 056 43 072 43 088
43217 43233 43 249
43 377 43 393 43 409
43 537 43 553 43 569
43696 43 712 43 727
43 854 43 870 43 886
44 012 44 028 44 044
44 170 44 185 44201
44326 44342 44358
44483 44498 44514
44638 44654 44669
44793 44809 44824
44948 44963 44 979
45 102 45 117 45 ^Z?,
45 255 45 271 45 286
45 408 45 423 45 439
45561 45576 45591
45 712 45 728 45 743
45 864 45 879 45 894
46 oi5 46 030 46 045
46 165 46 180 46 195
46315 46330 46345
46 464 46 479 46 494
46 613 46 627 46 642
46 761 46 776 46 790
46 909 46 923 46 938
47 056 47 070 47 085
47 202 47 217 47 232
47 349 47 363 47 378
47 494 47 509 47 524
47 640 47 654 47 669
47 784 47 799 47 813
39 933
40 106
40 278
40449
40 620
40790
40 960
41 128
41 296
41 464
41 631
41 797
41 963
42 127
42 292
42455
42619
42 781
42943
43 104
43 265
43425
43584
43 743
43 902
44059
44217
44 373
44529
44685
44840
44 994
45 148
45301
45 454
45 606
45 758
45 909
46 060
46 210
46359
46509
46657
46 805
46953
47 100
47 246
47392
47538
47683
47828
8
39950
40 123
40 295
40 466
40637
40 807
40976
41 145
41 313
41 481
41 647
41 814
41 979
42 144
42 308
42 472
42 635
42 797
42959
43 120
43 281
43441
43 600
43 759
43917
44075
44232
44389
44545
44700
44855
45 010
45 163
45 317
45469
45 621
45 773
45 924
46 075
46 225
46374
46532
46 672
46 820
46967
47 114
47 261
47 407
47 553
47698
47842
9
N
12 3 4
5 6 7 8 9
D
300
47 712 47 727 47 741 47 75^ 47 77©
47 784 47 799 47 813 47 828 47 842
14+
301
47857 47871 47885 47900 47914
47 929 47 943 47 958 47 972 47 986
14
302
48 001 48 015 48 029 48 044. 48 058
48 073 48 087 48 loi 48 116 48 130
14
303
48 144 48 159 48 173 48 187 48 202
48 216 48 230 48 244 48 259 48 273
14
304
48287 48302 48316 48330 48344
48359 48373 48387 48401 48416
14
305
48 430 48 444 48 458 48 473 48 487
48501 48515 48530 48544 48558
14
306
48572 48586 48601 48615 48629
48643 48657 48671 48686 48700
14
307
48 714 48 728 48 742 48 756 48 770
48785 48799 48813 48827 48841
14
308
48855 48869 4888s 48897 48 911
48 926 48 940 48 954 48 968 48 982
14
309
48996 49010 49024 49038 49052
49 066 49 080 49 094 49 108 49 122
14
310
49 136 49 150 49 164 49 ^78 49 192
49 206 49 220 49 234 49 248 49 262
14
311
49276 49290 49304 49318 49332
49346 49360 49 374 49388 49402
14
312
49415 49429 49 443 49457 49471
49485 49 499 49513 49527 49541
14
3'^3
49 554 49568 49582 49596 49610
49624 49638 49651 49665 49679
14
314
49 693 49 707 49 721 49 734 49 748
49 762 49 776. 49 790 49 803 49 817
14
315
49831 49845 49859 49872 49886
49900 49 9U 49927 49941 49955
14
3^6
49969 49982 49996 50010 50024
50037 50051 50065 50079 50092
14
317
50 106 50 120 50 133 50 147 50 161
50174 50188 50202 50215 50229
14
3i«
50243 50256 50270 50284 50297
50 311 50325 50338 50352 50365
14
319
50379 50393 50406 50420 50433
50447 50461 50474 50488 50501
14
320
50515 50529 50542 50556 50569
50583 50596 50610 50623 50637
14
321
50651 50664 50678 50691 50705
50718 50732 50745 50759 50772
13+
322
50786 50799 50813 50826 50840
50853 50866 50880 50893 50907
13+
323
50920 50934 50947 50961 50974
50987 51 001 51 014 51028 51 041
13
324
51 055 51 068 51 081 51 095 51 108
51 121 51 135 51 148 51 162 51 175
13
325
51 188 51 202 51 215 51 228 51 242
51 255 51 268 51 282 51 295 51 308
13
326
51322 51335 51348 51362 51375
51388 51402 51 415 51428 51 441
13
327
51455 51468 51 481 51495 51508
51 521 51534 51548 51 561 51574
13
328
51 587 51 601 51 614 51 627 51 640
51 654 51 667 51 680 51 693 51 706
13
329
51720 51 733 51 746 51759 51 772
51 786 51 799 51 812 51 825 51 838
13
330
51 851 51865 51878 51 891 51904
51917 51930 5x943 51957 51970
13
331
51983 51996 52009 52022 52035
52 048 52 061 52 075 52 088 52 lOI
13
332
52 114 52 127 52 140 52 153 52 166
52 179 52 192 52 205 52 218 52 231
13
333
52 244 52 257 52 270 52 284 52 297
52310 52323 52336 52349 52362
13
334
52 375 52 388 52 401 52 414 52 427
52440 52453 52466 52479 52492
13
335
52504 52517 52530 52543 52556
52569 52582 52595 52608 52621
13
33(^
52 634 52 647 52 660 52 673 52 686
52 699 52 711 52 724 52 737 52 750
13
337
52 763 52 776 52 789 52 802 52 815
52827 52840 52853 52866 52879
^3'
338
52892 52905 52917 52930 52943
52956 52969 52982 52994 53007,
13
339
53020 53033 53046 53058 53071
53 084 53 097 53 "o 53 122 53 135
13
340
53 148 53 161 53 173 53 186 53 199
53 212 53 224 53 237 53 250 53 263
13
341
53275 53288 53301 53314 53326
53 339 53352 53364 53377 53390
13
342
53403 53 4^5 53428 53441 53453
53466 53479 53491 53504 53517
13
343
53529 53542 53555 53567 53580
53 593 53605 53618 53631 53643
13
344
53 656 53 668 53 681 53 694 53 706
53 719 53 732 53 744 53 757 53 769
13
345
53 782 53 794 53 807 53 820 53 832
53845 53857 53870 53882 53895
13
346
53908 53920 53 933 53 945 53 958
53970 53983 53 995 54008 54020
12+
347
54033 54045 54058 54070 54083
54095 54108 54120 54133 54145
12+
348
54158 54170 54183 54195 54208
54220 54233 54245 54258 54270
124-
349
54283 54295 54307 54320 54332
54 345 54 357 54 37© 54382 54 394
12
350
54407 54419 54432 54 444 54456
54469 54481 54494 54506 54518
12
N
1 2 3 4
5 6 7 8 9
N
12 3 4
5 6 7 8 9
1)
350
54407 54 4^9 54432 54 444 54 45^
54469 54481 54494 54506 54518
12
351
54531 54 543 54 555 54 568 54580
54 593 54605 54617 54630 54642
12
352
54654 54667 54679 54691 54704
54716 54728 54741 54 753 54765
12
353
54777 54790 54802 54814 54827
54839 54851 54864 54.876 54888
12
354
54900 54913 54925 54 937 54 949
54962 54 974 54986 54998 55 on
12
355
55 023 55 035 55 047 55 060 55 072
55084 55096 55 108 55 1.21 55 133
12
356
55 145 55 157 55 ^^9 55 182 55 194
55206 55218 55230 55242 55 255.
12
357
55267 55279 55291 55303 55315
55328 55340 55352 55364 55376
12
358
55 Z^^ SS 400 55 413 55 425 55 437 '
55 449 55 461 55 473 55 485 55 497
12
359
55509 55522 55534 55546 55558
55570 55582 55594 55606 55618
12
360
55 630 55 642 55 654 55 666 55 678
55 691 55 703 55 715 55 727 55 739
12
361
55751 55763 55 775 55 787 55 799
55 811 55823 55835 55847 55859
12
362
55871 55883 55895 55907 55919
55931 55 943 55955 55967 55 979
12
363
55991 56003 56015 56027 56038
56050 56062 56074 56086 56098
12
364
56 no 56 122 56 134 56 146 56 158
56 170 56 182 56 194 5*6 205 56 217
12
365
56 229 56 241 56 253 56 265 56 277
56289 56 301 56 312 56324 56336
12
366
56348 56360 56372 56384 56396
56407 56419 56431 56443 56455
12
367
56467 56478 56490 56502 56514
56526 56538 56549 56561 56573
12
368
56585 56597 56608 56620 56632
56644 56656 56667 56679 56691
12
369
56703 56714 56726 56738 56750
56761 56773 56785 56797 56808
12
370
56820 56832 56844 56855 56867
56879 56891 56902 56914 56926
12
371
56937 56949 56961 56972 56984
56996 57008 57019 57031 57043
12
372
57 054 57 066 57 078 57 089 57 loi
57 113 57 124 57 136 57 148 57 159
12
373
57 171 57 ^^2, 57 i94 57 206 57 217
57 229 57 241 57 252 57 264 57 276
12
374
57287 57299 57310 57322 57334
57 345 57 357 57 368 57380 57 392
12
375
57 403 57 415 57 426 57 438 57 449
57461 57 473 57484 57496 57507
12
376
57519 57530 57542 57 553 57565
57576 57588 57600 57 611 57623
11+
377
57 634 57 646 57 657 57 669 57 680
57 692 57 703 57 715 57 726 57 738
"+
378
57 749 57 761 57 772 57 784 57 795
57807 57818 57830 57841 57852
"+
379
57864 57875 57887 57898 57910
57921 57 933 57 944 57 955 57 967
380
57978 57990 58001 58013 58024
58035 58047 58058 58070 58081
381
58092 58 104 58 115 58 127 58 138
58 149 58 161 58 172 58 184 58 195
382
58 206 58 218 58 229 58 240 58 252
58263 58274 58286 58297 58309
383
583^0 58331 58343 58354 58365
58377 58388 58399 58410 58422
384
58433 58444 58456 58467 58478
58490 58501 58512 58524 58535
385
58546 58557 58569 58580 58591
58602 58614 58625 58636 58647
386
58659 58670 58681 58692 58704
58715 58726 58737 58749 58760
387
58771 58782 58794 58805 58816
58827 58838 58850 58861 58872
388
58883 58894 58906 58917 58928
58939 58950 58961 58973 58984
389
58995 59006 59017 59028 59040
59051 59062 59073 59084 59095
390
59 106 59 118 59 129 59 140 59 151
59162 59173 59184 59195 59207
391
59218 59229 59240 59251 59262
59273 59284 59295 59306 59318
392
59329 59340 59351 59362 59373
59384 59 395 59406 59417 59428
393
59439 59450 59461 59472 59483
59494 59506 59517 59528 59 539
394
59550 59561 59572 59583 59594
59605 59616 59627 59638 59649
395
59660 59671 59682 59693 59704
59 715 59 726 59 737 59 748 59 759
396
59770 59780 59791 59802 59813
59824 59835 59846 59857 59868
397
59879 59890 59901 59912 59923
59 934 59945 59956 59966 59 977
398
59988 59999 60010 60021 60032
60 043 60 054 60 065 60 076 60 086
399
60 097 60 108 60 119 60 130 60 141
60 152 60 163 60 173 60 184 60 195
400
60206 60217 60228 60239 60249
60260 60271 60282 60293 60304
N
12 3 4
5 6 7 8 9
N
12 3 4
5 6 7 8
9
»
400
60 206 60 217 60 228 60 239 60 249
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60 412
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60 520
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60 627
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60853 6086^ 60874 60885 60895
60906 60917 60927 60938
60 949
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407
60959 60970 60981 609.91 61002
61 013 61 023 61 034 61 045
61 055
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61 066 61 077 61 087 61 098 61 109
61 119 61 130 61 140 61 151
61 162
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61 225 61 236 61 247 61 257
61 268
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61 278 61 289 61 300 61 310 61 321
61 331 61342 61 352 61 s6s
61 374
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61 384 61 395 61 405 61 416 61 426
61 437 61 448 61 458 6x 469
61 479
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61 542 61 553 61 563 61 574
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61 595 61 606 61 616 61 627 61 637
61 648 61 658 61 669 61 679
61 690
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61 794
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61 899
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61 909 61 920 61 930 61 941 61 951
61 962 61 972 61 982 61 993
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62 014 62 024 62 034 62 045 62 055
62 066 62 076 62 086 62 097
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62 170 62 180 62 190 62 201
62 211
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62 377 62 387 62 397 62 408
62 418
10
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62 480 62 490 62 500 62 511
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62 583 62 593 62 603 62 613
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62 634 62 644 62 655 62 665 62 675
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62 788 62 798 62 808 62 8i8
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62 839 62 849 62 859 62 870 62 880
62 890 62 900 62 910 62 921
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63 337
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63347 63357 6s 367 63377 63387
63397 63407 63417 63428 63438
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63 448 63 458 6s 468 6s 478 63 488
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63 548 63 558 6s 568 63 579 63 589
63 599 63 609 63 619 63 629
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63 749 63 759 63 769 6s 779 63 789
63 799 63 809 63 819 63 829
63839
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6s 849 63 859 6s 869 6s 879 6s 889
63 899 6s 909 6s 919 6s 929
63939
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436
63 949 63 959 63 969 63 979 6s 988
63998 64008 64018 64028
64038
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437
64048 64058 64068 64078 64088
64098 64108 64 118 64128
64 137
10
438
64147 64157 64167 64177 64187
64197 64207 64217 64227
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439
64246 64256 64266 64276 64286
64296 64306 64316 64326
64335
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440
64345 64355 64365 64375 64385
64395 64404 64414 64424
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64444 64454 64464 64473 64483
64493 64503 64513 64523
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442
64542 64552 64562 64572 64582
64591 64601 64 611 64621
64631
10
443
64 640 64 650 64 660 64 670 64 680
64689 64699 64709 64719
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64738 64748 64758 64768 64777
64787 64797 64807 64816
64826
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445
64836 64846 64856 64865 64875
64885 64895 64904 64914
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446
64933 64943 64953 64963 64972
64982 64992 65002 65 on
65 021
10
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65 °3i 65 040 65 050 6^ 060 65 070
65 079 65 089 65 099 65 108
65 118
10
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65 128 65 137 65 147 65 157 65 167
65 176 65 186 65 196 65 205 65 2l5
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65 225 65 234 65 244 65 254 65 263
65 273 65 283 65 292 65 302
65312
10
450
65321 65331 65341 65350 65360
65 369 65 379 65 389 65 398 65 408
10
N
12 3 4
5 6 7 8
9
N
12 3 4
5 6 7 8 9
D
450
65 321 65 331 65 341 65 350 65 360
65 369 65 379 65 389 65 398 65 408
10
451
65 418 65 427 65 437 65 447 65 456
65 466 65 475 65 485 65 495 65 504
10
452
65 514 65 523 65 533 65 543 65 552
65562 65571 65581 65591 65600
10
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65 610 65 619 65 629 65 639 65 648
65 658 65 667 65 677 65 686 65 696
10
454
65 706 65 715 65 725 65 734 65 744
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10
455
65 801 65 811 65 820 65 830 65 839
65 849 65 858 65 868 65 877 65 887
9+
456
65 896 65 906 65 916 65 925 65 935
65 944 65 954 65 963 65 973 65 982
9+
457
65992 66001 66 on 66020 66030
66039 66049 66058 66068 66077
9+
458
66 087 66 096 66 106 66 ii5 66 124
66 134 66 143 66 153 66 162 66 172
9+
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66 181 66 191 66200 66210 66219
66229 66238 66247 66257 66266
9+
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66276 66285 66295 66304 66314
66323 66332 66342 66351 66361
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66370 66 sSo 66389 66398 66408
66417 66427 664^6 66445 66455
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66 464 66 474 66 483 66 492 66 502
66 511 66521 66530 66539 26549
9
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66558 66567 66577 66586 66596
66605 66614 66624 66 6^^ 66642
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66652 66661 66671 666S0 66689
66699 66708 66717 66727 66736
9
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66 745 66 755 66 764 66 773 66 783
66792 66801 66 811 66820 66829
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66 8^9 66848 66857 66867 66876
66885 66894 66904 66913 66922
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467
66 932 66 941 66 950 66 960 66 969
66978 66987 66997 67006 67015
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468
67025 67034 67043 67052 67062
67 071 67 080 66 089 67 099 67 108
9
469
67 117 67 127 67 136 67 145 67 154
67 164 67 173 67 182 67 191 67 201
9
470
67 210 67 219 67 228 67 237 67 247
67 256 67 265 67 274 67 284 67 293
9
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67302 67 311 67321 67330 67339
67348 67357 67367 67376 67385
9
472
67394 67403 67413 67422 67431
67 440 67 449 67 459 67 468 67 477
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67 486 67 495 67 504 67 514 67 523
67 532 67 541 67 550 67 560 67 569
9
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67578 67587 67596 67605 67614
67624 6"] 6;^^ 67642 67651 67660
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475
67 669 67 679 67 688 67 697 67 706
67715 67724 67733 67742 67752
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67 761 67 770 67 779 67 788 67 797
67 806 67 815 67 825 67 834 67 843
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477
67 852 67 861 67 870 67 879 67 888
67897 67906 67916 67925 67934
9
478
67943 67952 67961 67970 67979
67988 67997 68006 68015 68024
9
479
68034 68043 68052 68061 68070
68079 68088 68097 68106 68 115
9
480
68 124 68 133 68 142 68 151 68 160
68 169 68 178 68 187 68 196 68 205
9
481
68215 68224 68233 68242 68251
68 260 68 269 68 278 68 287 68 296
9
482
68305 68314 68323 68332 68341
68350 68359 68368 68377 68386
9
483
68395 68404 68413 68422 68431
68 440 68 449 68 458 68 467 68 476
9
484
68485 68494 68502 68 511 68520
68529 68538 68547 68556 68565
9
485
68574 68583 68592 68601 68610
68619 68628 68637 68646 68655
9
486
68 664 68 673 68 681 68 690 68 699
68 708 68 717 68 726 68 735 68 744
9
487
68753 68762 68771 68780 68789
68797 68806 68815 68824 68833
9
488
68842 68851 68860 68869 68878
68 886 68895 68904 68913 68922
9
489
68931 68940 68949 68958 68966
68975 68984 68993 69002 69 01 1
9
490
69 020 69 028 69 037 69 046 69 055
69 064 69 073 69 082 69 090 69 099
9
491
69 108 69 117 69 126 69 135 69 144
69 152 69 161 69 170 69 179 69 188
9
492
69 197 69 205 69 214 69 223 69 232
69 241 69 249 69 258 69 267 69 276
9
493
69285 69294 69302 69 311 69320
69329 69338 69346 69355 69364
9
494
69373 69381 69390 69399 69408
69417 69425 69434 69443 69452
9
495
69461 69469 69478 69487 69496
69504 69513 69522 69531 69539
9
496
69548 69557 69566 69574 69583
69592 69601 69609 69618 69627
9
497
69636 69644 69653 69662 69671
69679 69 688 69 697 69705 69714
9
498
69 723 69 732 69 740 69 749 69 758
69 767 69 715 69 784 69 793 69 801
9
499
69810 69819 69827 69836 69845
69854 69862 69871 69880 69888
9
500
69897 69906 69914 69923 69932
69 940 69 949 69 958 69 966 69 975
9
N
12 3 4
5 6 7 8 9
N
12 3 4
5 6 7 8 9
D
500
69 897 69 906 69 914 69 923 69 932
69 940 69 949 69 958 69 966 69 975
9
501
69984 69992 70001 70010 70018
70027 70036 70044 70053 70062
9
502 1 70070 70079 70088 70096 70105
70 114 70122 70 131 70140 70148
9
503 70157 70165 70174 70183 70 191
70200 70209 70217 70226 70234
9
504
70243 70252 70260 70269 70278
70286 70295 70303 70312 70321
9
505
70329 70338 70346 70355 70364
70372 70381 70389 70398 70406
9
506
70415 70424 70432 70441 70449
70458 70467 70475 70484 70492
9
507
70501 70509 70518 70526 70535
70544 70552 70561 70569 70578
9
508
70586 70595 70603 70612 70621
70629 70638 70646 70655 70663
«+
509
70672 70680 70689 70697 70706
70714 70725 70731 70740 70749
8+
510
70757 70766 70774 70783 70791
70800 70808 70817 70825 70834
8+
511
70842 70851 70859 70868 70876
70885 70893 70902 70910 70919
8+
512
70927 70935 70944 70952 70961
70969 70978 70986 70995 71003
8+
513
71 012 71020 71029 71037 71046
71054 71063 71 071 71079 71088
8+
514
71 096 71 105 71 113 71 122 71 130
71 139 71 147 71 155 71 164 71 172
8
515
71 181 71 189 71 198 71 206 71 214
71 223 71 231 71 240 71 248 71 257
8
516
71 265 71 273 71 282 71 290 71 299
71307 71 315 71324 71332 71 341
8
517
71349 71357 71366 71374 71383
71 391 71399 71408 71 416 71425
8
518
71433 71 441 71450 71458 71466
71475 71483 71492 71500 71508
8
519
71 517 71525 71533 71542 71550
71 559 71 567 71 575 71 584 71 592
8
520
71 600 71 609 71 617 71 625 71 634
71642 71650 71659 71667 71675
8
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71 684 71 692 71 700 71 709 71 717
71 725 71 734 71 742 71 750 71 759
8
522
71 767 71 775 71 784 71 792 71 800
71809 71 817 71825 71834 71842
8
523
71850 71858 71867 71875 71883
71892 71900 71908 71 917 71925
8
524
71933 71 941 71950 71958 71966
71975 71983 71 991 71999 72008
8
525
72 016 72 024 72 032 72 041 72 049
72057 72066 72074 72082 72090
8
526
72 099 72 107 72 115 72 123 72 132
72 140 72 148 72 156 72 165 72 173
8
527
72 181 72 189 72 198 72 206 72 214
72 222 72 230 72 239 72 247 72 255
8
528
72 263 72 272 72 280 72 288 72 296
72304 72313 72321 72329 72337
8
529
72346 72354 72362 72370 72378
72387 72395 72403 72 411 72419
8
530
72 428 72 436 72 444 72 452 72 460
72469 72477 72485 72493 72501
8
531
72509 72518 72526 72534 72542
72550 72558 72567 72575 72583
8
532
72591 72599 72607 72616 72624
72 632 72 640 72 648 72 656 72 665
8
533
72673 72681 72689 72697 72705
72 713 72 722 72 730 72 738 72 746
8
534
72 754 72 762 72 770 72 779 72 787
72795 72803 72 811 72819 72827
8
535
72835 72843 72852 72860 72868
72 876 72 884 72 892 72 900 72 908
8
536
72916 72925 72933 72941 72949
72957 72965 72973 72981 72989
8
537
72997 73006 73014 73022 73030
73038 73046 73054 73062 73070
8
538
73078 73086 73094 73102 73 III
73 "9 73 127 73 135 73 i43 73 151
8
539
73 159 73 167 73 175 73 183 73 191
73199 73207 73215 73223 73231
8
540
73 239 73 247 73 255 73 263 73 272
73280 73288 73296 73304 73312
8
541
73320 73328 73336 73 344 73352
73360 73368 73376 73384 73392
8
542
73 400 73 408 73 416 73 424 73 432
73 440 73 448 73 456 73 464 73 472
8
543
73480 73488 73496 73504 73512
73520 73528 73536 73 544 73552
8
544
73560 73568 73576 73584 73592
73 600 73 608 73 616 73 624 73 632
8
545
73 640 73 648 73 656 73 664 73 672
73679 73687 73695 73703 73 7"
8
546
73 719 73 7.27 J73 735 73 743 73 75^
73 759 73 767 73 775 73 783 73 79^
8
547'
73 799 73 807 73 8i5 73 823 73 830
73^3^ 73846 73854 73862 73870
8
548
73878 73^86 73894 73902 73910
73918 73926 73 933 73941 73 949
8
549
73 957 73965 73 973 73 981 73989
73997 74005 74013 74020 74028
8
550
74036 74044 74052 74060 74068
74076 74084 74092 74099 74107
8
5f
1 2 3 4
5 6 7 8 9
N
550
1
3 3
4
5 6 7 8 9
D
74036 74044
74052 74060
74068
74076 74084 74092 74099 74107
8
551
74 115 74123
74 131 74139
74147
74155 74162 74170 74178 74186
8
552
74194 74202
74 210 74 218
74225
74233 74241 74249 74257 74265
8
553
74273 74280
74 288 74 296
74304
74312 74320 74327 74335 74343
8
554
74351 74359
74367 74 374
74382
74390 74398 74406 74414 74421
8
555
74429 74 437
74445 74 453
74461
74468 74476 74484 74492 74500
8
556
74507 74515
74523 74531
74 539
74 547 74 554 74 562 74 57o 74 578
8
557
74586 74 593
74601 74609
74617
74624 74632 74640 74648 74656
8
558
74663 74671
74679 74687
74695
74702 74710 74718 74726 74 733
8
559
74741 74749
74 757 74764
74772
74780 74788 74796 74803 74 81 1
8
560
74819 74-827
74834 74842
74850
74858 74865 74873 74881 74889
8
561
74896 74904
74912 74920
74927
74935 74943 74950 74958 74966
8
562
74 974 74981
74989 74 997
75005
75 012 75 020 75 028 75 035 75 043
8
563
75051 75059
75 066 75 074
75082
75089 75097 75 io5 75 113 75 120
8
564
75 128 75 136
75 143 75 151
75 159
75 166 75 174 75 182 75 189 75 197
8
565
75 205 75 213
75 220 75 228
75236
75 243 75 251 75 259 75 266 75 274
8
566
75 282 75 289
75 297 75 305
75312
75320 75328 75335 75343 75351
8
567
75358 75366
75 374 75 381
75389
75 397 75404 75412 75420 75427
8
568
75 435 75 442
75 450 75 458 75 465
75 473 75 481 75 488 75 496 75 504
8
569
75 511 75519
75526 75 534
75542
75 549 75 557 75 565 75 572 75 580
8
570
75587 75 595
75 ^03 75 610
75618
75 626 75 633 75 641 75 648 75 656
8
571
75664 75671
75 679 75 686
75694
75 702 75 709 75 717 75 724 75 732
8
572
75 740 75 747
75 755 75 762
75 770
75 778 75 785 75 793 75 800 75 808
8
573
75815 75823
75831 75838 75846
75 853 75 861 75 868 75 876 75 884
8
574
75 891 75 899 75906 75 914
75921
75 929 75 937 75 944 75 952 75 959
8
575
75967 75 974
75982 75989
75 997
76005 76012 76020 76027 76035
7+
576
76 042 76 o5o
76 057 76 065 76 072
76080 76087 76095 76103 76 no
7+
577
76 118 76 125
76 133 76 140
76 148
76155 76163 76170 76178 76185
7+
578
76 193 76 200
76 208 76 215
76223
76 230 76 238 76 245 76 253 76 260
7+
579
76 268 76 275
76 283 76 290
76 298
76305 76313 76320 76328 76335
7+
580
76343 76350
76358 76365 76373
76380 76388 76395 76403 76410
7+
581
76 418 76 425
76433 76440
76448
76455 76462 76470 76477 76485
7+
582
76492 76500
76507 76515
76522
76530 76537 76545 76552.76559
7+
583
76567 7.6574
76582 76589 76597
76 604 76 612 76 619 76 626 76 634
584
76641 76649 76656 76664 76671
76 678 76 686 76 693 76 701 76 708
585
76 716 76 723
76 730 76 738 76 745
76753 76760 76768 76775 76782
586
76 790 76 797
76805 76812
76819
76827 76834 76842 76849 76856
587
76864 76871
76879 76886
76893
76901 76908 76916 76923 76930
588
76938 76945 76953 76960
76967
76975 76982 76989 76997 77004
589
77012 77019
77026 77034
77041
77048 77056 77063 77070 77078
590
77085 77093
77 100 77 107
77 115
77 122 77 129 77 137 77 144 77 151
591
77 159 77 166
77 173 77 181
77188
77195 77203 77210 77217 77225
592
77 232 77 240
77 247 77 254
77262
77269 77276 77283 77291 77298
593
77305 77313
77320 77327
77335
77342 77 349 77 357 77 364 77 37i
594
77 379 77386
77 393 77401
77408
77415 77422 77430 77337 77444
595
77452 77 459
77466 77 474
77481
77488 77495 77503 77510 77517
596
77525 77532
77 539 77546
77 554
77561 77568 77576 77583 77590
597
77 597 77 605
77612 77619
77627
77634 77641 77648 77656 77663
598
77670 77677
77 685 77 692
77699
77706 77714 77721 77728 77 735
599
77 743 77 75o
77 757 77 764
77772
77 779 77786 77793 77801 77808
600
77815 77822
77830 77837 77844
77851 77859 77866 77873 77880
7
N
1
2 3
4
5 6 7 8 9
N
1 2
3 4
5 6 7 8 9
1)
600
60 1
602
603
604
77815 77822 77830 77837 77844
77887 77895 77902 77909 77916
77960 77967 77974 77981 77988
78032 78039 78046 78053 78061
78104 78 III 78 118 78125 78132
77851 77859 77866 77873 77880
77924 77931 77938 77945 77952
77996 78003 78010 78017 78025
78068 78075 78082 78089 78097
78 140 78 147 78 154 78 161 78 168
605
606
607
608
609
78 176 78 183 78 190 78 197 78 204
78247 78254 78262 78269 78276
78319 78326 78333 78340 78347
78390 78398 78405 78412 78419
78 462 78 469 78 476 78 483 78 490
78 211 78219 78226 78233 78240
78283 78290 78297 78305 78312
78355 78362 78369 78376 78383
78426 78433 78440 78447 78455
78497 78504 78512 78519 78526
610
611
612
613
614
78533 78540 78547 78554 78561
78604 78 611 78618 78625 'jsess
78675 78682 78689 78696 78704
78746 78753 78760 78767 78774
78817 78824 78831 78838 78845
78569 78576 78583 78590 78597
78640 78647 78654 78661 78668
78 711 78718 78725 78732 78739
78781 78789 78796 78803 78810
78852 78859 78866 78873 78880
615
616
617
618
619
78 888 78 895 78 902
78958 78965 78972
79029 79036 79043
79099 79 106 79 113
79 169 79 176 79 ^^3
78 909 78 916
78979 78986
79050 79057
79 120 79 127
79 190 79 197
78923 78930 78937 78944 78951
78993 79000 79007 79014 79021
79064 79071 79078 79085 79092
79 134 79 141 79 148 79 ^55 79 162
79204 79 211 79218 79225 79232
620
621
622
623
624
79 239 79 246 79 253
79309 79316 79323
79 379 79386 79 393
79 449 79456 79463
79518 79525 79532
79 260^79 267
79330*79337
79400 79407
79470 79 477
79 539 79546
79274 79281 79288 79295 79302
79 344 79351 79358 79365 79372
79414 79421 79428 79435 79442
79484 79491 79498 79505 79 511
79 553 79560 79567 79 574 79 581
625
626
627
628
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79657 79664 79671
79727 79734 79741
79 796 79803 79810
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79 609 79 616
79678 79685
79 748 79 754
79817 79824
79886 79893
79623 79630 79637 79644 79650
79692 79699 79706 79713 79720
79 761 79 768 79 775 79 782 79 789
79831 79837 79844 79851 79858
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630
631
632
634
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80 106 80 113 80 120 80 127 80 134
80 175 80 182 80 188 80 195 80 202
80243 80250 80257 80264 80271
635
636
637
638
639
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80346 80353 80359
80414 80 421 80 428
80482 80489 80496
80550 80557 80564
80 298 80 305
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80 448 80 455 80 462 80 468 80 475
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640
641
642
643
644
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80686 80693 80699 80706 80713
80 754 80 760 80 767 80 774 80 781
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80889 80895 80902 80909 80916
80652 80659 80665 80672 80679
80 720 80 726 80 733 80 740 80 747
80787 80794 80801 80808 80814
80855 80862 80868 80875 80882
80922 80929 80936 80943 80949
645
646
647
648
649
80 956 80 963 80 969
81 023 81 030 81 037
81 090 81 097 81 104
81 158 81 164 81 171
81 224 81 231 81 238
80 976 80 983
81 043 81 050
81 III 81 117
81 178 81 184
81 245 81 251
80990 80996 81003 81 010 81 017
81 057 81 064 81 070 81 077 81 084
81 124 81 131 81 137 81 144 81 151
81 191 81 198 81 204 81 211 81 218
81 258 81 265 81 271 81 278 81 285
650
81 291 81 298 81 305
81 311 81 318
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650
651
652
653
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81358 81365 81 371 81378 81385
81 425 81 431 81 438 81 445 81 451
81 491 81 498 81 505 81 511 81 518
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81 391 81398 81405 81411 81418
81 458 81 465 81 471 81 478 81 485
81 525 81 531 81 538 81 544 81 551
81 591 81 598 81 604 81 611 81 617
655
656
657
658
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81 690 81 697 81 704 81 710 81 717
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81 889 81 895 81 902 81 908 81 915
81 657 81 664 81 671 81 677 81 684
81 723 81 730 81 737 81 743 81 750
81 790 81 796 81 803 81 809 81 816
81 856 81 862 81 869 81 875 81 882
81 921 81 928 81 935 81 941 81 948
660
661
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82 151 82 158 82 164 82 171 82 178
82 217 82 223 82 230 82 236 82 243
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82 249 82 256 82 263 82 269 82 276
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82 315 82 321 82 328 82 334 82 341
82 380 82 387 82 393 82 400 82 406
82445 82452 82458 82465 82471
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82 640 82 646 82 653 82 659 82 666
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83 187 83 193 83 200 83 206 83.213
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83 155 83 161 83 168 83 174 83 181
83 219 83 225 83 232 83 238 83 245
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83 696 83 702 83 708 83 715 83 721
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83 822 83 828 83 835 83 841 83 847
83 601 83 607 83 613 83 620 83 626
83 664 83 670 83 677 83 683 83 689
S3 727 83 734 83 740 83 746 83 753
83 790 83 797 83 803 83 809 83 816
83 853 S3 860 83 866 83 872 83 879
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6
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691
692
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84 136 84 142 84 148 84 155 84 i6i
83 916 83 923 83 929 83 935 83 942
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84042 84048 84055 84061 84067
84105 84 III 84 117 84123 84130
84 167 84 173 84 180 84 186 84 192
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6
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695
696
697
698
699
84198 84205 84 211 84217 84223
84261 84267 84273 84280 84286
84323 84330 84336 84342 84348
84386 84392 84398 84404 84410
84448 84454 84460 84466 84473
84 230 84 236 84 242 84 248 84 255
84292 84298 84305 84311 84317
84354 84361 84367 84373 84379
84417 84423 84429 84435 84442
84479 84485 84491 84497 84504
6
6
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6
6
700
84510 84516 84522 84528 84535.
84541 84547 84553 84559 84566
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700
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701
84572 84578 84584 84590 84597
84603 84609 84615 84621 84628
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702
84634 84640 84646 84652 84658
84665 84671 84677 84683 84689
6
703
84696 84702 84708 84714 84720
84726 84733 84739 84745 84751
6
704
84757 84763 84770 84776 84782
84788 84794 84800 84807 84813
6
705
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706
84880 84887 84893 84899 84905
84 911 84917 84924 84930 84936
6
707
84942 84948 84954 84960 84967
84973 84979 84985 84991 84997
6
708
85 003 85 009 85 016 85 022 85 028
85 034 85 040 85 046 85 052 85 058
6
709
85 065 85 071 85 077 85 083 85 089
85 095 85 loi 85 107 85 114 85 120
6
710
85 126 85 132 85 138 85 144 85 150
85 156 85 163 85 169 85 175 85 181
6
711
85 187 85 193 85 199 85 205 85 211
85 217 85 224 85 230 85 236 85 242
6
712
85 248 85 254 85 260 85 266 85 272
85 278 85 285 85 291 85 297 85 303
6
713
85309 85315 85321 85327 85333
85 339 85 345 85 352 85 358 85 364
6
714
85 370 85 376 85 382 85 388 85 394
85 400 85 406 85 412 85 418 85 425
6
715
85 431 85 437 85 443 85 449 85 455
85 461 85 467 85 473 85 479 85 485
6
716
85 491 85 497 85 503 85 509 85 516
85 522 85 528 85 534 85 540 85 546
6
717
85552 85558 85564 85570 85576
85 582 85 588 85 594 85 600 85 606
6
718
85 612 85 618 85 625 85 631 85 637
85 643 85 649 85 655 85 661 85 667
6
719
85 673 85 679 85 685 85 691 85 697
85 703 85 709 85 715 85 721 85 727
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85 733 85 739 85 745 85 751 85 757
85 763 85 769 85 775 85 781 85 788
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85 794 85 800 85 806 85 812 85 818
85 824 85 830 85 836 85 842 85 848
6
722
85 854 85 860 85 866 85 872 85 878
85 884 85 890 85 896 85 902 85 908
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723
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85 944 85 950 85 956 85 962 85 968
6
724
85 974 85 980 85 986 85 992 85 998
86004 86010 86016 86022 86028
6
725
86034 86040 86046 86052 86058
86064 86070 86076 86082 86088
6
726
86 094 86 100 86 106 86 112 86 118
86 124 86 130 86 136 86 141 86 147
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727
86 153 86 159 86 165 86 171 86 177
86 183 86 189 86 195 86 201 86 207
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728
86213 86219 86225 86231 86237
86 243 86 249 86 255 86 261 86 267
6
729
86273 86279 86285 86291 86297
86303 86308 86314 86320 86326
6
730
86332 86338 86344 86350 86356
86362 86368 86374 86380 86386
6
731
86392 86398 86404 86410 86415
86421 86427 86433 86439 86445
6
732
86451 86457 86463 86469 86475
86481 86487 86493 86499 86504
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86658 86664 86670 86676 86682
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736
86 688 86 694 86 700 86 705 86 711
86 717 86 723 86 729 86 735 86 741
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737
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738
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739
86864 86870 86876 86882 86 888
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6
740
86923 86929 86935 86941 86947
86953 86958 86964 86970 86976
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741
86 982 86 988 86 994 86 999 87 005
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742
87 040 87 046 87 052 87 058 87 064
87 070 87 075 87 081 87 087 87 093
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743
87 099 87 105 87 III 87 116 87 122
87 128 87 134 87 140 87 146 87 151
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744
87 157 87 163 87 169 87 17s 87 181
87 186 87 192 87 198 87 204 87 210
6
745
87 216 87 221 87 227 87 233 87 239
87 245 87 251 87 256 87 262 87 268
6
746
87 274 87 280 87 286 87 291 87 297
87303 87309 87315 87320 87326
6
747
87332 87338 87344 87349 87355
87361 87367 87373 87379 87384
6
748
87 390 87 396 87 402 87 408 87 413
87419 87425 87431 87437 87442
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749
87 448 87 454 87 460 8y 466 87 471
87 477 87 483 87 489 87 495 87 500
6
750
87506 87512 87518 87523 87529
87535 87541 87547 87552 87558
6
N
12 3 4
5 6 7 8 9
N
12 3 4
5 6 7 8 9
D
750
87506 87512 87518 87523 87529
87535 87541 87547 87552 87558
6
751
87564 87570 87576 87581 87587
87 593 87 599 87 604 87 610 87 616
6
752
87 622 87 628 87 633 87 639 87 645
87651 87656 87662 87668 87674
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753
87679 87685 87691 87697 87703
87 708 87 714 87 720 87 726 87 731
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754
87 737 87 743 87 749 87 754 87 760
87766 87772 87777 87783 87789
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755
87795 87800 87806 87812 87818
87823 87829 87835 87841 87846
6
756
87 852 87 858 87 864 87 869 87 875
87881 87887 87892 87898 87904
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757
87910 87915 87921 87927 87933
87 938 87 944 87 950 87 955 87 961
6
758
87 967 87 973 87 978 87 984 87 990
87996 88001 88007 88013 88018
6
759
88 024 88 030 88 036 88 041 88 047
88 053 88 058 88 064 88 070 88 076
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760
88081 88087 88093 88098 88104
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761
88 138 88 144 88 150 88 156 88 161
88 167 88 173 88 178 88 184 88 190
6
762
88195 88201 88207 88213 88218
88 224 88 230 88 235 88 241 88 247
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763
88 252 88 258 88 264 88 270 88 275
88281 88287 88292 88298 88304
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764
88309 88315 88321 88326 88332
88 338 88 343 88 349 88 355 88 360
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765
88366 88372 88377 88383 88389
88395 88400 88406 88412 88417
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766
88 423 88 429 88 434 88 440 88 446
88451 88457 88463 88468 88474
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767
88480 88485 88491 88497 88502
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6
768
88536 88542 88547 88553 88559
88564 88570 88576 88581 88587
6
769
88593 88598 88604 88610 88615
88621 88627 88632 88638 88643
6
770
88 649 88 655 88 660 88 666 88 672
88 677 88 683 88 689 88 694 88 700
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771
88705 88 711 88717 88722 88728
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772
88762 88767 88773 88779 88784 1 88790 88795 88801 88807 88812
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773
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774
88874 88880 88885 88891 88897 88902 88908 88913 88919 88925
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775
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776
88 986 88 992 88 997 89 003 89 009
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777
89 042 89 048 89 053 89 059 89 064
89070 89076 89081 89087 89092
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778
89 098 89 104 89 109 89 ii5 89 120
89 126 89 131 89 137 89 143 89 148
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779
89 154 89 159 89 165 89 170 89 176
89 182 89 187 89 193 8g 198 89 204
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780
89209 89215 89221 89226 89232
89 237 89 243 89 248 89 254 89 260
6
781
89265 89271 89276 89282 89287
89293 89298 89304 89310 89315
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782
89321 89326 89332 89337 89343
89348 89354 89360 89365 89371
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783
89376 89382 89387 89393 89398 1 89404 89409 89415 89421 89426
5+
784
89432 89437 89443 89448 89454
89459 89465 89470 89476 89481
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785
89 487 89 492 89 498 89 504 89 509
89515 89520 89526 89531 89537
5+
7S6
89542 89548 89553 89559 89564
89570 89575 89581 89586 89592
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89597 89603 89609 89614 89620
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89 653 89 658 89 664 89 669 89 675
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89 763 89 768 89 774 89 779 89 785
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89 982 89 988 89 993 89 998 90 004
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90 146 90 151 90 157 90 162 90 168
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92 038 92 044 92 049 92 054 92 059
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92 075 92 080 92 085
92 091 92 096 92 loi 92 106
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92 195 92 200 92 205 92 210
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92 247 92 252 92 257 92 262
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92 283 92 288 92 293
92 298 92 304 92 309 92 314
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92 335 92 340 92 345
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92 402 92 407 92 412 92 418
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92 480
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92 505 92 511 92 516 92 521
92526
5
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92531
92536
92 542 92547 92552
92 557 92562 92567 92572
92578
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92583
92588
92 593 92 598 92 603
92 609 92 614 92 619 92 624
92 629
5
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92 645 92 650 92 655
92 660 92 665 92 670 92 675
92 681
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92 691
92 696 92 701 92 706
92 711 92 716 92 722 92 727
92732
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92 737
92 742
92 747 92 752 92 758
92 763 92 768 92 773 92 778
92 783
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92 788
92 793
92 799 92 804 92 809
92 814 92 819 92 824 92 829
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92 840
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92 850 92 855 92 860
92865 92870 92875 92881
92 886
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92 891
92 896
92 901 92 906 92 911
92 916 92 921 92 927 92 932
92937
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92942
92947
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92967 92973 92978 92983
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1
2 3 4
5 6 7 8
9
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12 3 4
5 6
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850
851
852
853
854
92 942 92 947 92 952 92 957 92 962
92 993 92 998 93 003 93 008 93 013
93 044 93 049 93 054 93 ©59 93 064
93 095 93 100 93 105 93 no 93 115
93 146 93 151 93 156 93 161 93 166
92 967 92973
93018 93024
93 069 93 075
93 120 93 125
93 171 93 176
92 978 92 983 92 988
93 029 93 034 93 039
93 080 93 085 93 090
93 131 93 136 93 141
93 181 93 186 93 192
5
5
5
5
5
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856
857
858
859
93 197 93 202 93 207 93 212 93 217
93 247 93 252 93 258 93 263 93 268
93298 93303 93308 93313 93318
93 349 93 354 93 359 93 3^4 93 3^9
93 399 93 304 93 409 93 4i4 93 420
93222 93 227
93 273 93 278
93323 93328
93 374 93 379
93 425 93 430
93 232 93 237 93 242
93 283 93 288 93 293
93 334 93 339 93 344
93 384 93 389 93 394
93 435 93 440 93 445
5
5
5
5
5
860
861
862
863
864
93 450 93 455 93 460 93 465 93 470
93500 93505 93510 93515 93520
93551 93556 93561 93566 93571
93601 93606 93 611 93616 93621
93651 93656 93661 93666 93671
93 475 93480
93526 93531
93576 93581
93 626 93 631
93 676 93 682
93 485 93 490 93 495
93 536 93 541 93 546
93586 93591 93596
93 63^ 93 641 93 646
93 687 93 692 93 697
5
5
5
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866
867
868
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93 702 93 707 93 712 93 717 93 722
93 752 93 757 93 762 93 767 93 772
93802 93807 93812 93817 93822
93852 93857 93862 93867 93872
93902 93907 93912 93917 93922
93 727 93 732
93 777 93 782
93827 93832
93877 93882
93927 93932
93 737 93 742 93 747
93 787 93 792 93 797
93 837 93 842 93 847
93 887 93 892 93 897
93 937 93942 93 947
5
5
5
5
5
870
871
872
873
874
93952 93 957 93962 93967 93972
94002 94007 94012 94017 94022
94052 94057 94062 94067 94072
94 loi 94106 94 III 94 116 94 121
94 151 94156 94 161 94166 94 171
93 977 93 982
94027 94032
94077 94082
94 126 94 131
94 176 94 181
93 987 93 992 93 997
94037 94042 94047
94086 94091 94096
94 136 94 141 94 146
94 186 94 191 94 196
5
5
5
5
5
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876
877
878
879
94201 94206 94 211 94216 94221
94250 94255 94260 94265 94270
94300 94305 943^0 94 3^5 94320
94 349 94 354 94 359 94364 94369
94 399 94404 94409 94414 94419
94226 94231 94236 94240 94245
94275 94280 94285 94290 94295
94325 94330 94335 94340 94 345
94374 94379 94384 94389 94 394
94 424 94 429 94 433 94 438 94 443
5
5
5
5
5
880
881
882
883
884
94448 94453 94458 94463 94468
94498 94503 94507 94512 94517
94 547 94552 94 557 94562 94567
94596 94601 94606 94 611 94616
94645 94650 94655 94660 94665
94 473 94478
94522 94527
94571 94576
94621 94626
94670 94675
94483 94488 94 493
94532 94 537 94542
94581 94586 94591
94630 94635 94640
94 680 94 685 94 689
5
5
5
5
5
885
886
887
888
889
94694 94699 94^04 94709 94714
94 743 94 748 94 753 94 758 94 763
94792 94797 94802 94807 94812
94841 94846 94851 94856 94861
94890 94895 94900 94905 94910
94719 94724 94729 94 734 94738
94768 94 773 94778 94783 94787
94817 94822 94827 94832 94836
94866 94871 94876 94880 94885
94915 94919 94924 94929 94 934
5
5
5
5
5
890
891
892
893
894
94 939 94944 94 949 94 954 94 959
94 988 94 993 94 998 95 002 95 007
95 036 95 041 95 046 95 051 95 056
95 085 95 090 95 095 95 100 95 105
95 134 95 139 95 143 95 148 95 i53
94963 94968 94 973 94978 94983
95012 95017 95022 95027 95032
95061 95066 95071 95075 95080
95 109 95 114 95 119 95 124 95 129
95 158 95 163 95 168 95 173 95 177
5
5
5
5
5
895
896
897
898
899
95 182 95 187 95 192 95 197 95 202
95 231 95 236 95 240 95 245 95 250
95 279 95 284 95 289 95 294 95 299
95 328 95 332 95 337 95 342 95 347
95 376 95 381 95 386 95 390 95 395
95 207 95 211 95 216 95 221 95 226
95 255 95 260 95 265 95 270 95 274
95303 95308 95313 95318 95323
95352 95357 95361 95366 95371
95 400 95 405 95 410 95 415 95 419
5
5
5
5
5
900
95 424 95 429 95 434 95 439 95 444
95 448 95 453
95 458 95 463 95 468
5
N
12 3 4
5 6
7 8 9
N
12 3 4
5 6 7 8 9
D
900
95 424 95 429 95 434 95 439 95 444
95 448 95 453 95 458 95 463 95 468
5
901
95 472 95 477 95 482 95 487 95 492
95 497 95 501 95 506 95 511 95 5^6
5
902
95521 95525 95530 95535 95540
95 545 95 550 95 554 95 559 95 564
5
9°3
95 569 95 574 95 578 95 583 95 588
95 593 95 598 95 602 95 607 95 612
5
904
95 617 95 622 95 626 95 631 95 636
95 641 95 646 95 650 95 655 95 660
5
90s
95 665 95 670 95 ^74 95 679 95 684
95 689 95 694 95 698 95 703 95 708
5
906
95 713 95 718 95 722 95 727 95 732
95 737 95 742 95 746 95 751 95 756
5
907
95 761 95 766 95 770 95 775 95 780
95 785 95 789 95 794 95 799 95 804
5
908
95 809 95 813 95 818 95 823 95 828
95 832 95 837 95 842 95 847 95 852
5
909
95 856 95 861 95 866 95 871 95 875
95 880 95 885 95 890 95 895 95 899
5
910
95 904 95 909 95 914 95 9^8 95 923
95 928 95 933 95 938 95 942 95 947
5
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95952 95 957 95961 95966 95971
95 976 95 980 95 985 95 99o 95 995
5
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95 999 96 004 96 009 96 014 96 019
96 023 96 028 96 033 96 038 96 042
5
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96047 96052 96057 96061 96066
96071 96076 96080 96085 96090
5
914
96 095 96 099 96 104 96 109 96 114
96 118 96 123 96 128 96 133 96 137
5
915
96 142 96 147 96 152 96 156 96 161
96 166 96 171 96 175 96 180 96 185
5
916
96 190 96 194 96 199 96 204 96 209
96213 96218 96223 96227 96232
5
917
96237 96242 96246 96251 96256
96 261 96 265 96 270 96 275 96 280
5
918
96 284 96 289 96 294 96 298 96 303
96308 96313 96317 96322 96327
5
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96332 96336 96341 96346 96350
96355 96360 96365 96369 96374
5
920
96379 96384 9^3^^ 9^393 96398
96402 96407 96412 96417 96421
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96426 96431 96435 96440 96445
96 450 96 454 96 459 96 464 96 468
5
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96473 96478 96483 96487 96492
96497 96501 96506 96 511 96515
5
923
96520 96525 96530 96534 96539
96544 96548 96553 96558 96562
5
924
96567 96572 96577 96581 96586
96591 96595 96600 96605 96609
5
925
96614 96619 96624 96628 96633
96658 96642 96647 96652 96656
5
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96 661 96 666 96 670 96 675 96 680
96 685 96 689 96 694 96 699 96 703
5
927
96 708 96 713 96 717 96 722 96 727
96 731 96 736 96 741 96 745 96 750
5
928
96 755 96 759 96 764 96 769 96 774
96 778 96 783 96 788 96 792 96 797
5
929
96802 96806 96 811 96816 96820
96 825 96 830 96 834 96 839 96 844
5
930
96848 96853 96858 96862 96867
96872 96876 96881 96886 96890
5
931
96 895 96 900 96 904 96 909 96 914
96918 96923 96928 96932 96937
5
932
96942 96946 96951 96956 96960
96965 96970 96974 96979 96984
5
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96988 96993 96997 97002 97007
97 on 97016 97021 97025 97030
5
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97 035 97 039 97 044 97 o49 97 ^53
97058 97063 97067 97072 97077
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97 081 97 086 97 090 97 095 97 100
97 104 97 109 97 114 97 118 97 123
5
936
97 128 97 132 97 137 97 142 97 146
97 151 97 155 97 160 97 165 97 169
5
937
97 174 97 179 97 183 97 188 97 192
97 197 97 202 97 206 97 211 97 216
5
938
97 220 97 225 97 230 97 234 97 239
97 243 97 248 97 253 97 257 97 262
5
939
97 267 97 271 97 276 97 280 97 285
97 290 97 294 97 299 97 304 97 308
5
940
97313 97317 97322 97327 97331
97336 97340 97 345 97350 97 354
5
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97 359 97364 97368 97 373 97 377
97382 97387 97391 97396 97400
5
942
97405 97410 97414 97419 97424
97428 97 433 97 437 97 442 97447
5
943
97 451- 97 456 97 460 97 465 97 470
97 474 97 479 97 483 97 488 97 493
5
944
97 497 97502 97506 97 511 97 5^6
97520 97525 97529 97 534 97 539
5
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97 543 97548 97552 97 557 97562
97566 97571 97 575 97580 97585
5
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97 589 97 594 97 598 97 603 97 607
97 612 97 617 97 621 97 626 97 630
5
947
97 635 97 640 97 644 97 649 97 653
97658 97663 97667 97672 97676
5
948
97 681 97 685 97 690 97 695 97 699
97 704 97 708 97 713 97 717 97 722
5
949
97727 97731 97736 97740 97745
97 749 97 754 97 759 97 763 97 768
5
950
97772 97777 97782 97786 97791
97 795 97 800 97 804 97 809 97 813
5
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12 3 4
5 6 7 8 9
N
13 3 4
5 6 7 8 9
D
950
■97 772 97 777 97 782 97 786 97 791
97 795 97 800 97 804 97 809 97 813
5
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97818 97823 97827 97832 97836
97841 97845 97850 97855 97859
5
952
97 864 97 868 97 873 97 877 97 882
97 886 97 891 97 896 97 900 97 905
5
953
97 909 97 914 97 918 97 923 97 928
97932 97 937 97941 97946 97950
5
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97 955 97 959 97 9^4 97 9'^^ 97 973
97978 97982 97987 97991 97996
5
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98000 98005 98009 98014 98019
98 023 98 028 98 032 98 037 98 041
4+
956
98 046 98 050 98 055 98 059 98 064
98068 98073 98078 98082 98087
4+
957
98 091 98 096 98 100 98 io5 98 109
98 114 98 118 98 123 98 127 98 132
4+
95S
98 137 98 141 98 146 98 150 98 155
98 159 98 164 98 168 98 173 98 177
4+
959
98 182 98 186 98 191 98 195 98 200
98 204 98 209 98 214 98 218 98 223
44-
960
98 227 98 232 98 236 98 241 98 245
98 250 98 254 98 259 98 263 98 268
4+
961
98 272 98 277 98 281 98 286 98 290
98295 98299 98304 98308 98313
4+
962
98318 98322 98327 98331 98336
98340 98345 98349 98354 98358
4+
963
98363 98367 98372 98376 98381
98 385 98 Z9° 98 394 9^ 399 98 403
4+
964
98408 98412 98417 98421 98426
98 430 98 435 98 439 98 444 98 448
4+
965
98453 98457 98462 98466 98471
98 475 98 480 98 484 98 489 98 493
4+
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98498 98502 98507 9851T 98516
98520 98525 98529 98534 98538
4+
967
98543 98547 98552 98556 98561
98565 98570 98574 98579 98583
4+
968
98588 98592 98597 98601 98605
98610 98614 98619 98623 98628
4+
969
98632 98637 98641 98646 98650
98 655 98 659 98 664 98 668 98 673
4+
970
98677 98682 98686 98691 98695
98 700 98 704 98 709 98 713 98 717
4+
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98722 98726 98731 98735 98740
98 744 98 749 98 753 98 758 98 762
4+
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98 767 98 771 98 776 98 780 98 784
98 789 98 793 98 798 98 802 98 807
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98811 98816 98820 98825 98829
98834 98838,98843 98847 98851
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98856 98860 98865 98869 98874
98878 98883 98887 98892 98896
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98 900 98 905 98 909 98 914 98 918
98923 98927 98932 98936 98941
4+
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98 945 98 949 98 954 98 958 98 963
98967 98972 98976 98981 98985
4
977
98 989 98 994 98 998 99 003 99 007
99012 99016 99021 99025 99029
4
978
99034 99038 99043 99047 99052
99056 99061 99065 99069 99074
4
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99 078 99 083 99 087 99 092 99 096
99 100 99 io5 99 109 99 114 99 118
4
980
99 123 99 127 99 131 99 136 99 140
99 145 99 149 99 154 99 158 99 162
4
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99 167 99 171 99 176 99 180 99 185
99 189 99 193 99 198 99 202 99 207
4
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99 211 99 216 99 220 99 224 99 229
99 233 99 238 99 242 99 247 99 251
4
983
99 255 99 260 99 264 99 269 99 273
99277 99282 99286 99291 99295
4
984
99300 99304 99308 99313 99317
99322 99326 99330 99335 99ZZ9
4
98s
99 344 99348 99352 99357 99361
99366 99370 99 374 99 379 99383
4
986
99388 99392 99396 99401 99405
99410 99414 99419 99423 99427
4
987
99 432 99 436 99 441 99 445 99 449
99 454 99458 99463 99467 99471
4
988
99 476 99 480 99 484 99 489 99 493
99498 99502 99506 99 511 99515
4
989
99520 99524 99528 99533 99.537
99542 99546 99550 99555 99 559
4
990
99564 99568 99572 99577 99581
, 99 585 99 590 99 594 99 599 99 603
4
991
99607 99612 99616 99621 99625
99 629 99 634 99 638 99 642 99 647
4
992
99651 99656 99660 99664 99669
99673 99677 99682 99686 99691
4
993
99 695 99 699 99 704 99 708 99 712
99717 99721 99726 99730 99 734
4
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99 739 99 743 99 747 99 752 99 756
99 760 99 765 99 769 99 774 99 778
■ 4
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99782 99787 99791 99 795 99800
99804 99808 99813 99817 99822
4
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99826 99830 99835 99839 99843
99848 99852 99856 99861 99865
4
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99870 99874 99878 99883 99887
99891 99896 99900 99904 99909
4
998
99913 99917 99922 99926 99930
99 935 99 939 99 944 99 948 99 952
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999
99 957 99961 99965 99970 99974
99978 99983 99987 99991 99996
4
1000
00000 00004 00009 00013 00017
00 022 00 026 00 030 00 035 00 039
4
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12 3 4
5 6 7 8 9
Answers.
Page 4.
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Page 6.
1. 439
2. 141
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Page 12.
78.40
50.72
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Page 19.
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^
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