Multiplication
Methods
A collection of some methods for doing
multiplication that have been devised and used in
previous times.
Notes on Multiplication Methods ~ 1
All of the methods given here are written methods.
For an additional account of some mechanical methods see under 'Artifacts for
Mathematics' (in the trol menu) where the material for making, together with instructions for
using: Napier's Rods, Genaille's Rods and a Slide Rule, are to be found.
At the back of this unit are some sheets of multiplication tables.
The first is simply eight copies of a basic 1 to 9 multiplication square, which makes it easy
to provide everyone with their own personal copy when those facts are needed. Needing
to “know their tables” should not be made an issue in this work.
The second, ‘A Book of Multiplication Tables’ is much more complicated to make. First of
all it requires some care with the double-sided photocopying needed. Check that the crosses
in the middle of the two sheets match (within 2 mm if possible), and also that the 94× table
does back the 89× table (and NOT the 65×).
The sheet has then to be folded in half three times. Do not rush this stage. Press the folds
down well with some hard object (ruler or pen). The dotted lines are only a guide, the
important thing is to match up the edges of the paper each time, and see that the dotted
lines are on the inside of the fold. The titled front cover should always be on the outside
of the fold.
A staple through the last fold (the hinged edge) and three cuts along the outer edges with
a guillotine will complete the process. It is nowhere near as difficult as it sounds!
An alternative cover for the booklet is provided. This will have to be cut and pasted onto the
master copy before the photocopying is done.
All of this would only be worthwhile if the tables might be used in some other work.
These same tables are also available in a straight sheet format if wanted. They can be
found in the ‘Division’ unit (in the trol menu).
It must be emphasied that there are many more ways of doing multiplication than those
given here, though most of them are merely variations on a general theme. The ones
suggested in this unit are those considered to offer the most useful insights into the historic
development of this particular piece of arithmetic so some might appreciate that
“No, it hasn’t always been like that!”
Exactly how any or all of this material is used will be, as always, a matter for individual
teachers to decide and plan for.
It can be surprising to see the lengths to which mathematicians and other professional
users of mathematics have gone to over the centuries to simplify the multiplication process.
But remember that even so great a mathematician as Leibnitz (1646 to 1716) who invented
the calculus said
“It is unworthy of excellent men to lose hours
like slaves in the labour of calculation.”
(A feeling which many of our pupils share!)
© Frank Tapson 2004 [trolFG:2]
Notes on Multiplication Methods ~ 2
Using Doubling and Halving
1. The Ancient Egyptian Method
This is probably the oldest of all the written methods. It is of some interest because of
its links with the binary form of a number, and also because of the simplicity of the idea - it
needs only an ability to be able to multiply by 2 and add.
However it should also be appreciated that in those days (1000 BC or so) such things
were not for 'ordinary folk' who would have had no need of such a skill. It was very much the
preserve of professionals such as astrologers, architects and mathematicians.
2. The Russian Peasant Method
This method was more widely used than its name might imply. It was in fact taught and
practised throughout Europe for quite some time. It is a less demanding algorithm than the
previous one since the numbers to be crossed off are easier to identify. It is interesting to
write them side by side and see how they are doing the same thing. The link of course is
the binary system underlying both.
1
2
4
8
16
32
43
268
536
1072
2144
4288
8576
11524
43
21
10
5
2
1
268
536
1072
2144
4288
8576
11524
Using the Gelosia Method
(Pronounced jee - low - see - uh)
2
Any analysis or explanation of why this method works is probably best
linked to looking at the place values of the different figures involved.
6
8
The worked example from the sheet is shown on the right.
2
4
1
It is easy to extend the diagram to cover tens of thousands and upwards
or, tenths, hundredths, thousandths and so on downwards, and thus show
why placing the decimal point works as it does.
8
H
T
Th
Th
4
4
3
×
H
H
H
H
2
U
T
T
U
U
T
T
This method almost certainly originated in India, as did so much of our arithmetic.
From there it spread outwards to China, Persia and the Arab world. It reached Europe
through Italy where it first appeared in the 14th century.
The name 'gelosia' was given to it in Italy since the necessary grid resembled the
lattices or gratings (= gelosia) which were made in metal and fastened in front of the
windows of the houses for security.
It has been claimed that Napier got the idea for his rods from this method. Whether
or not he did, the possibility can clearly be seen. In fact, multiplication tables and a
knowledge of place value could be dispensed with by using the two methods together.
Once the grid has been drawn, a set of Napier's Rods would provide all the needed
answers to complete it, and the final addition would be the only arithmetic necessary.
One practical point. Do not slow things up by drawing precisely ruled grids. It is
quite sufficient to write down the two numbers (adequately spaced) and sketch the
grid around them free-hand.
© Frank Tapson 2004 [trolFG:3]
3
2
6
The lower diagram replaces each of the the figures in the top diagram with
its place value, using the usual notation of H T U, with the addition of Th for
thousands. The neat way each place value falls into line along the diagonals
makes it very clear why the method works. It also makes it very apparent why
zeros must be included in the numbers when necessary.
×
8
Notes on Multiplication Methods ~ 3
Using Quarter Squares
The reason why this method works can only be justified by the use of algebra.
It can be shown that
(a + b) 2
ab ≡
4
(a – b) 2
–
4
Given a and b are the two numbers to be multiplied together, and the two terms
on the right are quarter squares (as defined on the sheet) then the algorithm follows
from the above identity. And (provided the requisite tables are available) multiplication
has been replaced by one addition and two subtractions without need for any
multiplication facts to be known or found.
In passing it is of interest to consider (and prove) why
the remainder can be disregarded.
Note that this method is completely accurate in giving all the figures possible in the
final answer. Its big limitations are that
(a) a set of tables is needed and,
(b) the size of the numbers it can handle is constrained by the size of
the tables (which has to cover the sum of the two numbers being
multiplied).
The second limitation is not too bad. The sample sheet included here goes to 600,
so it is easy to visualize the smallish book which would be needed to reach 100,000.
That means an accuracy of 4 significant figures (or better) which should cover most
practical needs. Approximations allow the tables to be used for larger numbers.
The method was used (but how widely is not known) and the necessary tables to
enable this were published in the late 1800’s.
Using Logarithms
This is a much simplified version of what was once taught in the classrooms of every
secondary school. The words ‘characteristic’ and ‘mantissa’ do not appear, and only
numbers greater than 1 are dealt with to avoid that very troublesome negative
characteristic.
Also only 3-figure tables are used and the customary difference table does not
appear. However, the essentials required to give a glimpse of what was done are there.
In modern terms this could be called “Logarithms Lite”.
The characteristic which is so necessary to control the place value has been replaced
here by the e-notation. This is the notation introduced by the IEEE (= Institute of Electrical
and Electronic Engineers) and used in all computer work. Gaining an acquaintance with
this notation might be regarded as a valuable by-product of this piece of work.
One of the inadequancies of 3-figure tables is very apparent once the numbers go
beyond 5.0 when many of the logarithmic values are repeated. This cannot be avoided
without moving on to 4 or 5-figure tables.
Sheet 1 does not require any knowledge of the characteristic as the examples are
chosen so as not to go beyond the range of the table. Also there are no ambiguous
values for the logarithms.
Sheet 2 introduces the idea of the e-notation.
Some help might be needed with the business of using the tables, first in finding the
logarithm from the number, and then the much more difficult matter of reversing that
process.
Further work could be done with numbers less than 1 and division.
© Frank Tapson 2004 [trolFG:4]
Notes on Multiplication Methods ~ 4
Using Triangle Numbers
This method has the big advantage that it can be easily explained or justified diagrammatically,
without any recourse to algebra, except for the generalisation at the end. For this purpose it is
better to change the conventionally drawn triangle numbers (as equilateral triangles) into rightangled triangles. The appropriate drawing for T4 is shown on the right.
We will do the sum 8 × 3 using five diagrams.
1.
2.
3.
The sum 8 × 3 requires
the number of objects
making this rectangle to
be counted.
First make the triangle
number which matches
the longest edge.
In this case it is T8
Cut away the triangle
(shown in black) which is
above the size of the
rectangle.
In this case it is T5
8
5
3
3
8
8
8
5.
4.
Triangle removed was T5
or T (8 - 3) leaving this
incomplete rectangle
Complete the rectangle
with smallest possible
triangle (shown in black).
In this case T2 or T(3 - 1)
3
8
Physically we have shown that 8 × 3 = 36 –
15
+
3
= T8 –
T5
+
T2
= T8 –
T(8 – 3)
+
T(3 – 1)
Which generalises to
a × b = Ta –
T(a – b)
+
T(b – 1)
Rearranged for the algorithm
a × b = Ta +
T(b – 1)
–
T(a – b)
Another approach to using
triangle numbers is implied by
this diagram which leads to a
different algorithm.
3
8
a
b
a+b
a+b
a
b
a
b
A similar thing can be done using square numbers (or indeed
any polygon numbers).
The identity
(a + b)2 = a2 + b2 + 2ab
which is illustrated on the left can be rearranged to give ab =
and the required algorithm can be produced from that.
A table giving the values of square numbers Sn is available.
Though interesting, and illuminating, to see how these work they were never more
than mathematical curiosities, and (unlike the method of quarter squares) there appears
to be no evidence that they were ever used in any practical way. One advantage this
method does have over quarter squares is that, in this case the largest number in the
tables is also the largest possible multiplier.
© Frank Tapson 2004 [trolFG:5]
Multiplication using Doubling and Halving
1. The Ancient Egyptian Method
Consider the sum 268 × 43
First write out two columns of numbers side by side.
The left-hand column starts with 1.
The right-hand colum starts with the larger of the two numbers being multiplied (268).
The columns are formed by doubling each of the previous numbers, stopping only
when the figure in the left-hand column would become bigger than (or equal to) the
other number being multiplied (43).
The final result would be as shown on the right.
Take care to keep the columns lined up, both vertically and horizontally.
The next stage is to look at the left-hand column and find the numbers which are needed
to add up to the value of the smaller multiplier (43).
In this case the necessary set is 1 + 2 + 8 + 32 = 43
(Note that the bottom number should always be needed, and it
is best working backwards from that, subtracting as you go.)
Cross out all the numbers in that column not needed together with the numbers in the
right-hand column which are on the same level.
Finally, add up the numbers in the right-hand column which have not been crossed out.
That is the answer. The completed sum is shown on the right.
So, 268 × 43 = 11524
1
2
4
8
16
32
268
536
1072
2144
4288
8576
1
2
4
8
16
32
43
268
536
1072
2144
4288
8576
11524
Try these, using the Ancient Egyptian method.
26 × 314
41 × 61
71 × 213
55 × 97
32 × 104
2. The Russian Peasant Method
Consider the sum 268 × 43
First write out two columns of numbers side by side.
Each column starts with one of the two numbers to be multiplied. (43 and 268)
Make one column by doubling each of the previous numbers.
Make the other column by halving each of the previous numbers,
ignoring any remainders, and stopping at 1.
(It is best to make the halving column under the smaller of the two multipliers)
251 × 21
43
21
10
5
2
1
268
536
1072
2144
4288
8576
43
21
10
5
2
1
268
536
1072
2144
4288
8576
11524
The final result would be as shown on the right.
Take care to keep the columns lined up, both vertically and horizontally.
The next stage is to look at the halving column and find all the even numbers.
In this case they are 10 and 2
Cross out the even numbers in that column together with the numbers in the right-hand
column which are on the same level.
Finally, add up the numbers in the right-hand column which have not been crossed out.
That is the answer. The completed sum is shown on the right.
So, 268 × 43 = 11524
Try these, using the Russian Peasant method.
24 × 316
© Frank Tapson 2004 [trolFG:6]
30 × 54
63 × 81
128 × 231
71 × 483
327 × 45
Multiplication using the Gelosia Method
Consider the sum 268 × 43 (That is 3 digits × 2 digits)
Sketch a grid like that on the right, and place the two numbers as shown.
2
6
×
8
4
3
Next, do the six single-digit multiplications formed by matching the separate
numbers across the top with those on the right-hand edge.
In this case, it means doing the six sums shown on the right
and the answers to those multiplications are written in the
appropriate boxes of the grid for each number-pair.
2×4
2×3
6×4
6×3
8×4
8×3
2
6
8
No answer can have more than two digits. (tens and units)
Any tens (L.H. digit) are written above the diagonal line,
The units (R.H. digit) are written below the diagonal line. (It may be zero)
The completed grid should look like that on the right.
2
8
3
4
1
8
2
6
2
8
1
5
4
4
3
×
8
3
4
1
1
2
2
6
The final answer is produced by adding (the results of the multiplication only)
along the diagonals, working from right to left, and carrying figures as
necessary, and this is shown on the right.
×
2
4
4
3
2
6
8
2
4
So, 268 × 43 = 11524
Try these, using the gelosia method.
386 × 24
758 × 34
76 × 89
Decimal points are dealt with very simply.
Consider the sum 2.68 × 4.3
Work exactly as before with 268 × 43 and then put the
two decimal points in those numbers on the vertical and
horizontal lines respectively.
Follow those two lines into the grid until they meet, and
then, moving down the diagonal line from there, put the
decimal point in the answer.
All of this can be seen in the diagram on the right by following
the arrows.
So, 2.68 × 4.3 = 11.524
402 × 31
999 × 78
478 × 219
2
6
2
8
3
4
1
6
1
1
.
5
×
8
2
4
4
3
2
8
2
4
Try these, using the gelosia method.
First, by using the grids already created in the previous work, write down the answers to these:
3.86 × 24
75.8 × 3.4
7.6 × 89
40.2 × 31
9.99 × 78
47.8 × 2.19
7.32 × 18
508 × 3.7
0.617 × 3.8
0.059 × 23
0.074 × 0.038
and then do these:
12.3 × 4.6
© Frank Tapson 2004 [trolFG:7]
Multiplication using Quarter Squares
The sheet headed Values of Quarter Squares Qn is needed
The quarter square of a number, is the value of that number multiplied by itself
and then divided by 4, ignoring any remainder (if there is one).
Examples
The quarter square of 2 is: 2 × 2 ÷ 4 = 4 ÷ 4 = 1
3 is: 3 × 3 ÷ 4 = 9 ÷ 4 = 2 rem 1 which becomes 2
5 is: 5 × 5 ÷ 4 = 25 ÷ 4 = 6 rem 1 which becomes 6
and so on.
Quarter squares would be of little use for the purpose of simplifying
multiplication if they had to be worked out each time they were needed and
so a table is used to look up their values.
Look at the sheet: Values of Quarter Squares and confirm that the quarter
square of 413 is 42 642 (or 42,642 if preferred).
Having such a table enables us to change a multiplication sum into a much
simpler matter of addition and subtraction.
Consider the sum 268 × 43
First add them:
268 + 43 = 311 and find the quarter square of that = 24 180
Then subtract them: 268 – 43 = 255 and find the quarter square of that = 12 656
Finally, subtract the quarter squares to get the final answer
= 11 524
and 268 × 43 = 11 524
It could be set out something like this:
×
+
–
268
43
311
225
Qn
24 180
12 656 –
11 524
Try these, using the quarter squares method.
245 × 32
© Frank Tapson 2004 [trolFG:8]
286 × 45
364 × 123
74 × 36
376 × 21
443 × 157
Multiplication using Triangle Numbers
The sheet headed Values of Triangle Numbers Tn is needed
T1 = 1
T2 = 3
T3 = 6
T4 = 10
T5 = 15
Triangle numbers are those that can be made in the way shown above where
the first five are drawn.
The actual value of any triangle number can be worked out using
Tn = n × (n + 1) ÷ 2
Example
The value of the triangle number for 8 is: 8 × (8 + 1) ÷ 2 = 8 × 9 ÷ 2 = 36
Triangle Numbers would be of little use for the purpose of simplifying
multiplication if they had to be worked out each time they were needed and
so a table is used to look up their values.
Look at the sheet: Values of Triangle Numbers and confirm that the triangle
number corresponding to 413 is 85 491 (or 85,491 if preferred).
Having such a table enables us to change a multiplication sum into a much
simpler matter of addition and subtraction.
Consider the sum 268 × 43
Find the triangle number corresponding to the larger (268)
Subtract 1 from the smaller (43 - 1 = 42) and find its triangle number
Add those two triangle numbers
Subtract the two multiplying numbers (268 - 43 = 225)
and find the triangle number for that
Subtract that from the previous result to get the final answer
and 268 × 43 = 11 524
= 36 046
= 12 656
= 36 949
= 25 425
= 11 524
It could be set out something like this:
Tn
268
36 046
43 – 1 = 42
903 +
36 949
– 225
25 425 –
11 524
Try these, using the triangle numbers method.
213 × 87
© Frank Tapson 2004 [trolFG:9]
367 × 42
179 × 53
485 × 132
593 × 341
259 × 34
Multiplication using Logarithms ~ 1
The sheet headed Values of Logarithms Ln is needed
We will not concern ourselves here with what logarithms are, but only how to
use them for multiplication. (They can also be used for division.)
First of all, every number has a logarithm associated with it and that association
is unique. In other words, given any number it has only one logarithm and, just
as importantly, given any logarithm it can only represent one number.
We will start with a very simple example of how we use logarithms.
Consider the sum 2 × 4
To use the table of logarithms we need to see this as 2.00 × 4.00
From the table we can find that 2.00 has a logarithm of
and that 4.00 has a logarithm of
Then we ADD the two logarithms to get
That is the answer to the sum but, it is a logarithm, so we need to find
what number it represents.
Using the table (in reverse) shows that 0.903 is the logarithm of 8.00
So 2.00 × 4.00 = 8.00 (Hardly a surprise, but good to see it works.)
0.301
0.602
0.903
Now consider the much harder sum 2.15 × 3.62
We find their logarithms are 0.332 and 0.559 and these add to 0.891
The logarithm 0.891 matches the number 7.78
So 2.15 × 3.62 = 7.78
It could be set out something like this:
N
2.15
× 3.62
7.78
Ln
0.332
0.559 +
0.891
The full answer should have been 7.783 but it must be recognised
that the very simplified table we are using is not capable of that
sort of accuracy. We wish only to establish how logarithms are
used and, for most practical purposes, this accuracy is adequate.
Try doing these, using logarithms
2×3
© Frank Tapson 2004 [trolFG:10]
3×3
1.92 × 3.36
5.18 × 1.45
3.55 × 1.29
4.07 × 1.93
Multiplication using Logarithms ~ 2
Notice that the table of logarithms covers only numbers from 1.00 to 9.99 and that is true for all tables
of logarithms. For greater accuracy it might be 1.0000 to 9.9999 but the principle is the same. So now
we consider how to adapt all numbers to fit the tables and see how powerful this tool is in handling all
multplications.
For this we need to be familiar with a particular way of writing numbers which first sets them down as
a value between 1 and 10 and then puts enough × 10's after them so they would be restored to their
correct value.
For example 43 can be written as 4.3 × 10
268 can be written as 2.68 × 10 × 10
There is a special notation used for this which writes the above numbers as
4.3e1 and 2.68e2 respectively
where the number after the 'e' shows how many 10's are needed.
Or it can be thought of as the amount that the decimal point must be
moved to the right to restore the correct value.
Now when we look up the logarithms of 4.3 and 2.68 we replace the 0 on the front with the e-number
(if there is one).
So that the logarithm of 43 (log 4.3 = 0.633 with e = 1) becomes 1.633
and the logarithm of 268 (log 2.68 = 0.428 with e = 2) becomes 2.428
Then we add the two logarithms to get 4.061
Remembering this stands for a logarithm of 0.061 with e = 4
We next find 0.061 in the logarithm tables and see that it corresponds to the number 1.15
With e = 4 we must multiply by 10, 4 times
1.15 × 10 × 10 × 10 × 10 = 11500
More accurately 43 × 268 = 11524
But we can write 43 × 268 = 11500 (to 3 significant figures)
It could be set out something like this:
N
43
× 268
1.15
Ln
1.633
2.428 +
4.061
Try doing these, using logarithms
37 × 26
© Frank Tapson 2004 [trolFG:11]
45 × 78
274 × 63
517 × 842
65.4 × 173
3610 × 904
© Frank Tapson 2004 [trolFG:12]
Qn
0
1
2
4
6
9
12
16
20
25
30
36
42
49
56
64
72
81
90
100
110
121
132
144
156
169
182
196
210
225
240
256
272
289
306
324
342
361
380
400
420
441
462
484
506
529
552
576
600
625
n
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
96
97
98
99
100
91
92
93
94
95
86
87
88
89
90
81
82
83
84
85
76
77
78
79
80
71
72
73
74
75
66
67
68
69
70
61
62
63
64
65
56
57
58
59
60
51
52
53
54
55
n
2304
2352
2401
2450
2500
2070
2116
2162
2209
2256
1849
1892
1936
1980
2025
1640
1681
1722
1764
1806
1444
1482
1521
1560
1600
1260
1296
1332
1369
1406
1089
1122
1156
1190
1225
930
961
992
1024
1056
784
812
841
870
900
650
676
702
729
756
Qn
146
147
148
149
150
141
142
143
144
145
136
137
138
139
140
131
132
133
134
135
126
127
128
129
130
121
122
123
124
125
116
117
118
119
120
111
112
113
114
115
106
107
108
109
110
101
102
103
104
105
n
5329
5402
5476
5550
5625
4970
5041
5112
5184
5256
4624
4692
4761
4830
4900
4290
4356
4422
4489
4556
3969
4032
4096
4160
4225
3660
3721
3782
3844
3906
3364
3422
3481
3540
3600
3080
3136
3192
3249
3306
2809
2862
2916
2970
3025
2550
2601
2652
2704
2756
Qn
196
197
198
199
200
191
192
193
194
195
186
187
188
189
190
181
182
183
184
185
176
177
178
179
180
171
172
173
174
175
166
167
168
169
170
161
162
163
164
165
156
157
158
159
160
151
152
153
154
155
n
9604
9702
9801
9900
10 000
9120
9216
9312
9409
9506
8649
8742
8836
8930
9025
8190
8281
8372
8464
8556
7744
7832
7921
8010
8100
7310
7396
7482
7569
7656
6889
6972
7056
7140
7225
6480
6561
6642
6724
6806
6084
6162
6241
6320
6400
5700
5776
5852
5929
6006
Qn
246
247
248
249
250
241
242
243
244
245
236
237
238
239
240
231
232
233
234
235
226
227
228
229
230
221
222
223
224
225
216
217
218
219
220
211
212
213
214
215
206
207
208
209
210
201
202
203
204
205
n
100
201
302
404
506
210
321
432
544
656
15
15
15
15
15
14
14
14
14
15
13
14
14
14
14
13
13
13
13
13
129
252
376
500
625
520
641
762
884
006
924
042
161
280
400
340
456
572
689
806
12 769
12 882
12 996
13 110
13 225
12
12
12
12
12
11 664
11 772
11 881
11 990
12 100
11 130
11 236
11 342
11 449
11 556
10 609
10 712
10 816
10 920
11 025
10
10
10
10
10
Qn
296
297
298
299
300
291
292
293
294
295
286
287
288
289
290
281
282
283
284
285
276
277
278
279
280
271
272
273
274
275
266
267
268
269
270
261
262
263
264
265
256
257
258
259
260
251
252
253
254
255
n
21
22
22
22
22
21
21
21
21
21
20
20
20
20
21
19
19
20
20
20
19
19
19
19
19
18
18
18
18
18
17
17
17
18
18
17
17
17
17
17
16
16
16
16
16
15
15
16
16
16
904
052
201
350
500
170
316
462
609
756
449
592
736
880
025
740
881
022
164
306
044
182
321
460
600
360
496
632
769
906
689
822
956
090
225
030
161
292
424
556
384
512
641
770
900
750
876
002
129
256
Qn
346
347
348
349
350
341
342
343
344
345
336
337
338
339
340
331
332
333
334
335
326
327
328
329
330
321
322
323
324
325
316
317
318
319
320
311
312
313
314
315
306
307
308
309
310
301
302
303
304
305
n
Values of Quarter Squares
29
30
30
30
30
29
29
29
29
29
28
28
28
28
28
27
27
27
27
28
26
26
26
27
27
25
25
26
26
26
24
25
25
25
25
24
24
24
24
24
23
23
23
23
24
22
22
22
23
23
929
102
276
450
625
070
241
412
584
756
224
392
561
730
900
390
556
722
889
056
569
732
896
060
225
760
921
082
244
406
964
122
281
440
600
180
336
492
649
806
409
562
716
870
025
650
801
952
104
256
Qn
396
397
398
399
400
391
392
393
394
395
386
387
388
389
390
381
382
383
384
385
376
377
378
379
380
371
372
373
374
375
366
367
368
369
370
361
362
363
364
365
356
357
358
359
360
351
352
353
354
355
n
39
39
39
39
40
38
38
38
38
39
37
37
37
37
38
36
36
36
36
37
35
35
35
35
36
34
34
34
34
35
33
33
33
34
34
32
32
32
33
33
31
31
32
32
32
30
30
31
31
31
204
402
601
800
000
220
416
612
809
006
249
442
636
830
025
290
481
672
864
056
344
532
721
910
100
410
596
782
969
156
489
672
856
040
225
580
761
942
124
306
684
862
041
220
400
800
976
152
329
506
Qn
446
447
448
449
450
441
442
443
444
445
436
437
438
439
440
431
432
433
434
435
426
427
428
429
430
421
422
423
424
425
416
417
418
419
420
411
412
413
414
415
406
407
408
409
410
401
402
403
404
405
n
Q n for n = 1 to 600
49
49
50
50
50
48
48
49
49
49
47
47
47
48
48
46
46
46
47
47
45
45
45
46
46
44
44
44
44
45
43
43
43
43
44
42
42
42
42
43
41
41
41
41
42
40
40
40
40
41
729
952
176
400
625
620
841
062
284
506
524
742
961
180
400
440
656
872
089
306
369
582
796
010
225
310
521
732
944
156
264
472
681
890
100
230
436
642
849
056
209
412
616
820
025
200
401
602
804
006
Qn
496
497
498
499
500
491
492
493
494
495
486
487
488
489
490
481
482
483
484
485
476
477
478
479
480
471
472
473
474
475
466
467
468
469
470
461
462
463
464
465
456
457
458
459
460
451
452
453
454
455
n
61
61
62
62
62
60
60
60
61
61
59
59
59
59
60
57
58
58
58
58
56
56
57
57
57
55
55
55
56
56
54
54
54
54
55
53
53
53
53
54
51
52
52
52
52
50
51
51
51
51
504
752
001
250
500
270
516
762
009
256
049
292
536
780
025
840
081
322
564
806
644
882
121
360
600
460
696
932
169
406
289
522
756
990
225
130
361
592
824
056
984
212
441
670
900
850
076
302
529
756
Qn
546
547
548
549
550
541
542
543
544
545
536
537
538
539
540
531
532
533
534
535
526
527
528
529
530
521
522
523
524
525
516
517
518
519
520
511
512
513
514
515
506
507
508
509
510
501
502
503
504
505
n
74
74
75
75
75
73
73
73
73
74
71
72
72
72
72
70
70
71
71
71
69
69
69
69
70
67
68
68
68
68
66
66
67
67
67
65
65
65
66
66
64
64
64
64
65
62
63
63
63
63
529
802
076
350
625
170
441
712
984
256
824
092
361
630
900
490
756
022
289
556
169
432
696
960
225
860
121
382
644
906
564
822
081
340
600
280
536
792
049
306
009
262
516
770
025
750
001
252
504
756
Qn
596
597
598
599
600
591
592
593
594
595
586
587
588
589
590
581
582
583
584
585
576
577
578
579
580
571
572
573
574
575
566
567
568
569
570
561
562
563
564
565
556
557
558
559
560
551
552
553
554
555
n
88
89
89
89
90
87
87
87
88
88
85
86
86
86
87
84
84
84
85
85
82
83
83
83
84
81
81
82
82
82
80
80
80
80
81
78
78
79
79
79
77
77
77
78
78
75
76
76
76
77
804
102
401
700
000
320
616
912
209
506
849
142
436
730
025
390
681
972
264
556
944
232
521
810
100
510
796
082
369
656
089
372
656
940
225
680
961
242
524
806
284
562
841
120
400
900
176
452
729
006
Qn
© Frank Tapson 2004 [trolFG:13]
136
153
171
190
210
231
253
276
300
325
351
378
406
435
465
496
528
561
595
630
666
703
741
780
820
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41 861
42 903
43 946
44 990
45 1035
1081
1128
1176
1225
1275
91
92
93
94
95
66
78
91
105
120
11
12
13
14
15
46
47
48
49
50
86
87
88
89
90
21
28
36
45
55
6
7
8
9
10
96
97
98
99
100
81
82
83
84
85
76
77
78
79
80
71
72
73
74
75
66
67
68
69
70
61
62
63
64
65
56
57
58
59
60
51
52
53
54
55
1
3
6
10
15
1
2
3
4
5
n
Tn
n
4656
4753
4851
4950
5050
4186
4278
4371
4465
4560
3741
3828
3916
4005
4095
3321
3403
3486
3570
3655
2926
3003
3081
3160
3240
2556
2628
2701
2775
2850
2211
2278
2346
2415
2485
1891
1953
2016
2080
2145
1596
1653
1711
1770
1830
1326
1378
1431
1485
1540
Tn
146
147
148
149
150
141
142
143
144
145
136
137
138
139
140
131
132
133
134
135
126
127
128
129
130
121
122
123
124
125
116
117
118
119
120
111
112
113
114
115
106
107
108
109
110
101
102
103
104
105
n
10 731
10 878
11 026
11 175
11 325
10 011
10 153
10 296
10 440
10 585
9316
9453
9591
9730
9870
8646
8778
8911
9045
9180
8001
8128
8256
8385
8515
7381
7503
7626
7750
7875
6786
6903
7021
7140
7260
6216
6328
6441
6555
6670
5671
5778
5886
5995
6105
5151
5253
5356
5460
5565
Tn
196
197
198
199
200
191
192
193
194
195
186
187
188
189
190
181
182
183
184
185
176
177
178
179
180
171
172
173
174
175
166
167
168
169
170
161
162
163
164
165
156
157
158
159
160
151
152
153
154
155
n
706
878
051
225
400
861
028
196
365
535
041
203
366
530
695
246
403
561
720
880
391
578
766
955
145
471
653
836
020
205
19
19
19
19
20
306
503
701
900
100
18 336
18 528
18 721
18 915
19 110
17
17
17
17
18
16
16
16
17
17
15 576
15 753
15 931
16 110
16 290
14
14
15
15
15
13
14
14
14
14
13
13
13
13
13
12
12
12
12
12
11 476
11 628
11 781
11 935
12 090
Tn
246
247
248
249
250
241
242
243
244
245
236
237
238
239
240
231
232
233
234
235
226
227
228
229
230
221
222
223
224
225
216
217
218
219
220
211
212
213
214
215
206
207
208
209
210
201
202
203
204
205
n
30
30
30
31
31
29
29
29
29
30
27
28
28
28
28
26
27
27
27
27
25
25
26
26
26
24
24
24
25
25
23
23
23
24
24
22
22
22
23
23
21
21
21
21
22
381
628
876
125
375
161
403
646
890
135
966
203
441
680
920
796
028
261
495
730
651
878
106
335
565
531
753
976
200
425
436
653
871
090
310
366
578
791
005
220
321
528
736
945
155
20 301
20 503
20 706
20 910
21 115
Tn
296
297
298
299
300
291
292
293
294
295
286
287
288
289
290
281
282
283
284
285
276
277
278
279
280
271
272
273
274
275
266
267
268
269
270
261
262
263
264
265
256
257
258
259
260
251
252
253
254
255
n
626
878
131
385
640
191
453
716
980
245
43
44
44
44
45
42
42
43
43
43
41
41
41
41
42
39
39
40
40
40
38
38
38
39
39
36
37
37
37
37
956
253
551
850
150
486
778
071
365
660
041
328
616
905
195
621
903
186
470
755
226
503
781
060
340
856
128
401
675
950
35 511
35 778
36 046
36 315
36 585
34
34
34
34
35
32 896
33 153
33 411
33 670
33 930
31
31
32
32
32
Tn
48
48
49
49
49
50
50
50
51
51
51
52
52
52
52
53
53
53
54
54
54 946
55 278
55 611
55 945
56 280
56
56
57
57
57
58 311
58 653
58 996
59 340
59 685
60
60
60
61
61
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
031
378
726
075
425
616
953
291
630
970
301
628
956
285
615
681
003
326
650
975
086
403
721
040
360
516
828
141
455
770
971
278
586
895
205
46
47
47
47
48
306
307
308
309
310
451
753
056
360
665
45
45
46
46
46
Tn
396
397
398
399
400
391
392
393
394
395
386
387
388
389
390
381
382
383
384
385
376
377
378
379
380
371
372
373
374
375
366
367
368
369
370
361
362
363
364
365
356
357
358
359
360
351
352
353
354
355
n
78
79
79
79
80
76
77
77
77
78
74
75
75
75
76
72
73
73
73
74
70
71
71
72
72
69
69
69
70
70
67
67
67
68
68
65
65
66
66
66
63
63
64
64
64
61
62
62
62
63
606
003
401
800
200
636
028
421
815
210
691
078
466
855
245
771
153
536
920
305
876
253
631
010
390
006
378
751
125
500
161
528
896
265
635
341
703
066
430
795
546
903
261
620
980
776
128
481
835
190
Tn
446
447
448
449
450
441
442
443
444
445
436
437
438
439
440
431
432
433
434
435
426
427
428
429
430
421
422
423
424
425
416
417
418
419
420
411
412
413
414
415
406
407
408
409
410
401
402
403
404
405
n
99
100
100
101
101
97
97
98
98
99
95
95
96
96
97
93
93
93
94
94
90
91
91
92
92
88
89
89
90
90
86
87
87
87
88
84
85
85
85
86
82
83
83
83
84
80
81
81
81
82
Tn for n = 1 to 600
301
302
303
304
305
n
Values of Triangle Numbers
681
128
576
025
475
461
903
346
790
235
266
703
141
580
020
096
528
961
395
830
951
378
806
235
665
831
253
676
100
525
736
153
571
990
410
666
078
491
905
320
621
028
436
845
255
601
003
406
810
215
Tn
926
378
831
285
740
106
106
107
107
108
491
953
416
880
345
104 196
104 653
105 111
105 570
106 030
101
102
102
103
103
Tn
496
497
498
499
500
491
492
493
494
495
526
003
481
960
440
123
123
124
124
125
120
121
121
122
122
256
753
251
750
250
786
278
771
265
760
341
828
316
805
295
115 921
116 403
116 886
117 370
117 855
113
114
114
114
115
111 156
111 628
112 101
112 575
113 050
486 118
487 118
488 119
489 119
490 120
481
482
483
484
485
476
477
478
479
480
471
472
473
474
475
466 108 811
467 109 278
468 109 746
469 110 215
470 110 685
461
462
463
464
465
456
457
458
459
460
451
452
453
454
455
n
546
547
548
549
550
541
542
543
544
545
536
537
538
539
540
531
532
533
534
535
526
527
528
529
530
521
522
523
524
525
516
517
518
519
520
511
512
513
514
515
506
507
508
509
510
501
502
503
504
505
n
601
128
656
185
715
981
503
026
550
075
386
903
421
940
460
816
328
841
355
870
271
778
286
795
305
751
253
756
260
765
916
453
991
530
070
149
149
150
150
151
331
878
426
975
525
146 611
147 153
147 696
148 240
148 785
143
144
144
145
146
141 246
141 778
142 311
142 845
143 380
138
139
139
140
140
135
136
137
137
138
133
133
134
134
135
130
131
131
132
132
128
128
129
129
130
125
126
126
127
127
Tn
596
597
598
599
600
591
592
593
594
595
586
587
588
589
590
581
582
583
584
585
576
577
578
579
580
571
572
573
574
575
566
567
568
569
570
561
562
563
564
565
556
557
558
559
560
551
552
553
554
555
n
177
178
179
179
180
174
175
176
176
177
171
172
173
173
174
169
169
170
170
171
166
166
167
167
168
163
163
164
165
165
160
161
161
162
162
157
158
158
159
159
154
155
155
156
157
152
152
153
153
154
906
503
101
700
300
936
528
121
715
310
991
578
166
755
345
071
653
236
820
405
176
753
331
910
490
306
878
451
025
600
461
028
596
165
735
641
203
766
330
895
846
403
961
520
080
076
628
181
735
290
Tn
© Frank Tapson 2004 [trolFG:14]
51
52
53
54
55
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
625
676
729
784
841
900
961
1024
1089
1156
1225
1296
1369
1444
1521
1600
1681
1764
1849
1936
2025
2116
2209
2304
2401
2500
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
8281
8464
8649
8836
9025
7396
7569
7744
7921
8100
6561
6724
6889
7056
7225
5776
5929
6084
6241
6400
5041
5184
5329
5476
5625
4356
4489
4624
4761
4900
3721
3844
3969
4096
4225
3136
3249
3364
3481
3600
2601
2704
2809
2916
3025
Sn
96 9216
97 9409
98 9604
99 9801
100 10 000
91
92
93
94
95
86
87
88
89
90
81
82
83
84
85
76
77
78
79
80
71
72
73
74
75
66
67
68
69
70
61
62
63
64
65
56
57
58
59
60
n
n Sn
146
147
148
149
150
141
142
143
144
145
136
137
138
139
140
131
132
133
134
135
126
127
128
129
130
121
122
123
124
125
116
117
118
119
120
111
112
113
114
115
106
107
108
109
110
101
102
103
104
105
n
21
21
21
22
22
19
20
20
20
21
18
18
19
19
19
17
17
17
17
18
15
16
16
16
16
14
14
15
15
15
13
13
13
14
14
12
12
12
12
13
316
609
904
201
500
881
164
449
736
025
496
769
044
321
600
161
424
689
956
225
876
129
384
641
900
641
884
129
376
625
456
689
924
161
400
321
544
769
996
225
11 236
11 449
11 664
11 881
12 100
10 201
10 404
10 609
10 816
11 025
Sn
196
197
198
199
200
191
192
193
194
195
186
187
188
189
190
181
182
183
184
185
176
177
178
179
180
171
172
173
174
175
166
167
168
169
170
161
162
163
164
165
156
157
158
159
160
151
152
153
154
155
n
38
38
39
39
40
36
36
37
37
38
34
34
35
35
36
32
33
33
33
34
30
31
31
32
32
29
29
29
30
30
27
27
28
28
28
25
26
26
26
27
24
24
24
25
25
22
23
23
23
24
416
809
204
601
000
481
864
249
636
025
596
969
344
721
100
761
124
489
856
225
976
329
684
041
400
241
584
929
276
625
556
889
224
561
900
921
244
569
896
225
336
649
964
281
600
801
104
409
716
025
Sn
246
247
248
249
250
241
242
243
244
245
236
237
238
239
240
231
232
233
234
235
226
227
228
229
230
221
222
223
224
225
216
217
218
219
220
211
212
213
214
215
206
207
208
209
210
201
202
203
204
205
n
60
61
61
62
62
58
58
59
59
60
55
56
56
57
57
53
53
54
54
55
51
51
51
52
52
48
49
49
50
50
46
47
47
47
48
44
44
45
45
46
42
42
43
43
44
40
40
41
41
42
516
009
504
001
500
081
564
049
536
025
696
169
644
121
600
361
824
289
756
225
076
529
984
441
900
841
284
729
176
625
656
089
524
961
400
521
944
369
796
225
436
849
264
681
100
401
804
209
616
025
Sn
296
297
298
299
300
291
292
293
294
295
286
287
288
289
290
281
282
283
284
285
276
277
278
279
280
271
272
273
274
275
266
267
268
269
270
261
262
263
264
265
256
257
258
259
260
251
252
253
254
255
n
87
88
88
89
90
84
85
85
86
87
81
82
82
83
84
78
79
80
80
81
76
76
77
77
78
73
73
74
75
75
70
71
71
72
72
68
68
69
69
70
65
66
66
67
67
63
63
64
64
65
616
209
804
401
000
681
264
849
436
025
796
369
944
521
100
961
524
089
656
225
176
729
284
841
400
441
984
529
076
625
756
289
824
361
900
121
644
169
696
225
536
049
564
081
600
001
504
009
516
025
Sn
346
347
348
349
350
341
342
343
344
345
336
337
338
339
340
331
332
333
334
335
326
327
328
329
330
321
322
323
324
325
316
317
318
319
320
311
312
313
314
315
306
307
308
309
310
301
302
303
304
305
n
96
97
97
98
99
93
94
94
95
96
90
91
91
92
93
276
929
584
241
900
041
684
329
976
625
856
489
124
761
400
721
344
969
596
225
636
249
864
481
100
601
204
809
416
025
Sn
119
120
121
121
122
116
116
117
118
119
112
113
114
114
115
716
409
104
801
500
281
964
649
336
025
896
569
244
921
600
396
397
398
399
400
391
392
393
394
395
386
387
388
389
390
381
382
383
384
385
376
377
378
379
380
371
372
373
374
375
366
367
368
369
370
361
362
363
364
365
356
357
358
359
360
351
352
353
354
355
n
156
157
158
159
160
152
153
154
155
156
148
149
150
151
152
145
145
146
147
148
141
142
142
143
144
137
138
139
139
140
133
134
135
136
136
130
131
131
132
133
126
127
128
128
129
123
123
124
125
126
816
609
404
201
000
881
664
449
236
025
996
769
544
321
100
161
924
689
456
225
376
129
884
641
400
641
384
129
876
625
956
689
424
161
900
321
044
769
496
225
736
449
164
881
600
201
904
609
316
025
Sn
446
447
448
449
450
441
442
443
444
445
436
437
438
439
440
431
432
433
434
435
426
427
428
429
430
421
422
423
424
425
416
417
418
419
420
411
412
413
414
415
406
407
408
409
410
401
402
403
404
405
n
Sn for n = 1 to 600
109 561
110 224
110 889
111 556
112 225
106
106
107
108
108
103
103
104
104
105
99
100
101
101
102
Values of Square Numbers
198
199
200
201
202
194
195
196
197
198
190
190
191
192
193
185
186
187
188
189
181
182
183
184
184
177
178
178
179
180
173
173
174
175
176
168
169
170
171
172
164
165
166
167
168
160
161
162
163
164
916
809
704
601
500
481
364
249
136
025
096
969
844
721
600
761
624
489
356
225
476
329
184
041
900
241
084
929
776
625
056
889
724
561
400
921
744
569
396
225
836
649
464
281
100
801
604
409
216
025
Sn
496
497
498
499
500
491
492
493
494
495
486
487
488
489
490
481
482
483
484
485
476
477
478
479
480
471
472
473
474
475
466
467
468
469
470
461
462
463
464
465
456
457
458
459
460
451
452
453
454
455
n
246
247
248
249
250
241
242
243
244
245
236
237
238
239
240
231
232
233
234
235
226
227
228
229
230
221
222
223
224
225
217
218
219
219
220
212
213
214
215
216
016
009
004
001
000
081
064
049
036
025
196
169
144
121
100
361
324
289
256
225
576
529
484
441
400
841
784
729
676
625
156
089
024
961
900
521
444
369
296
225
207 936
208 849
209 764
210 681
211 600
203 401
204 304
205 209
206 116
207 025
Sn
546
547
548
549
550
541
542
543
544
545
536
537
538
539
540
531
532
533
534
535
526
527
528
529
530
521
522
523
524
525
516
517
518
519
520
511
512
513
514
515
506
507
508
509
510
501
502
503
504
505
n
681
764
849
936
025
296
369
444
521
600
961
024
089
156
225
676
729
784
841
900
441
484
529
576
625
256
289
324
361
400
121
144
169
196
225
036
049
064
081
100
001
004
009
016
025
298 116
299 209
300 304
301 401
302 500
292
293
294
295
297
287
288
289
290
291
281
283
284
285
286
276
277
278
279
280
271
272
273
274
275
266
267
268
269
270
261
262
263
264
265
256
257
258
259
260
251
252
253
254
255
Sn
596
597
598
599
600
591
592
593
594
595
586
587
588
589
590
581
582
583
584
585
576
577
578
579
580
571
572
573
574
575
566
567
568
569
570
561
562
563
564
565
556
557
558
559
560
551
552
553
554
555
n
355
356
357
358
360
349
350
351
352
354
343
344
345
346
348
337
338
339
341
342
331
332
334
335
336
326
327
328
329
330
320
321
322
323
324
314
315
316
318
319
309
310
311
312
313
303
304
305
306
308
216
409
604
801
000
281
464
649
836
025
396
569
744
921
100
561
724
889
056
225
776
929
084
241
400
041
184
329
476
625
356
489
624
761
900
721
844
969
096
225
136
249
364
481
600
601
704
809
916
025
Sn
© Frank Tapson 2004 [trolFG:15]
0.176
0.204
0.230
0.255
0.279
0.301
0.322
0.342
0.362
0.380
0.398
0.415
0.431
0.447
0.462
0.477
0.491
0.505
0.519
0.531
0.544
0.556
0.568
0.580
0.591
0.602
0.613
0.623
0.633
0.643
0.653
0.663
0.672
0.681
0.690
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
0.654
0.664
0.673
0.682
0.691
0.603
0.614
0.624
0.634
0.644
0.545
0.558
0.569
0.581
0.592
0.479
0.493
0.507
0.520
0.533
0.400
0.417
0.433
0.449
0.464
0.303
0.324
0.344
0.364
0.382
0.179
0.207
0.233
0.258
0.281
1
0.004
0.045
0.083
0.117
0.149
0.655
0.665
0.674
0.683
0.692
0.604
0.615
0.625
0.635
0.645
0.547
0.559
0.571
0.582
0.593
0.480
0.494
0.508
0.521
0.534
0.401
0.418
0.435
0.450
0.465
0.305
0.326
0.346
0.365
0.384
0.182
0.210
0.236
0.260
0.283
2
0.009
0.049
0.086
0.121
0.152
0.656
0.666
0.675
0.684
0.693
0.605
0.616
0.626
0.636
0.646
0.548
0.560
0.572
0.583
0.594
0.481
0.496
0.509
0.522
0.535
0.403
0.420
0.436
0.452
0.467
0.308
0.328
0.348
0.367
0.386
0.185
0.212
0.238
0.262
0.286
3
0.013
0.053
0.090
0.124
0.155
0.657
0.667
0.676
0.685
0.694
0.606
0.617
0.627
0.637
0.647
0.549
0.561
0.573
0.584
0.596
0.483
0.497
0.510
0.524
0.537
0.405
0.422
0.438
0.453
0.468
0.310
0.330
0.350
0.369
0.387
0.188
0.215
0.241
0.265
0.288
4
0.017
0.057
0.093
0.127
0.158
0.658
0.667
0.677
0.686
0.695
0.607
0.618
0.628
0.638
0.648
0.550
0.562
0.574
0.585
0.597
0.484
0.498
0.512
0.525
0.538
0.407
0.423
0.439
0.455
0.470
0.312
0.332
0.352
0.371
0.389
0.190
0.217
0.243
0.267
0.290
5
0.021
0.061
0.097
0.130
0.161
0.659
0.668
0.678
0.687
0.695
0.609
0.619
0.629
0.639
0.649
0.551
0.563
0.575
0.587
0.598
0.486
0.500
0.513
0.526
0.539
0.408
0.425
0.441
0.456
0.471
0.314
0.334
0.354
0.373
0.391
0.193
0.220
0.245
0.270
0.292
6
0.025
0.064
0.100
0.134
0.164
0.660
0.669
0.679
0.688
0.696
0.610
0.620
0.630
0.640
0.650
0.553
0.565
0.576
0.588
0.599
0.487
0.501
0.515
0.528
0.540
0.410
0.427
0.442
0.458
0.473
0.316
0.336
0.356
0.375
0.393
0.196
0.223
0.248
0.272
0.294
7
0.029
0.068
0.104
0.137
0.167
0.661
0.670
0.679
0.688
0.697
0.611
0.621
0.631
0.641
0.651
0.554
0.566
0.577
0.589
0.600
0.489
0.502
0.516
0.529
0.542
0.412
0.428
0.444
0.459
0.474
0.318
0.338
0.358
0.377
0.394
0.199
0.225
0.250
0.274
0.297
8
0.033
0.072
0.107
0.140
0.170
To find the value of the logarithm of n
Match the first two digits of n (containing the decimal point) with
those in the left-hand column of the table. Go along that line of the
table to the three figure decimal number which lies directly under the
value at the head of the table matching the third digit of n.
That will be the logarithm of n.
0
0.000
0.041
0.079
0.114
0.146
1.0
1.1
1.2
1.3
1.4
0.662
0.671
0.680
0.689
0.698
0.612
0.622
0.632
0.642
0.652
0.555
0.567
0.579
0.590
0.601
0.490
0.504
0.517
0.530
0.543
0.413
0.430
0.446
0.461
0.476
0.320
0.340
0.360
0.378
0.396
0.201
0.228
0.253
0.276
0.299
9
0.037
0.076
0.111
0.143
0.173
Values of Logarithms
0
0.929
0.935
0.940
0.944
0.949
0.954
0.959
0.964
0.968
0.973
0.978
0.982
0.987
0.991
0.996
9.0
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
0.903
0.908
0.914
0.919
0.924
0.875
0.881
0.886
0.892
0.898
0.845
0.851
0.857
0.863
0.869
0.813
0.820
0.826
0.833
0.839
0.778
0.785
0.792
0.799
0.806
0.740
0.748
0.756
0.763
0.771
0.699
0.708
0.716
0.724
0.732
8.5
8.6
8.7
8.8
8.9
8.0
8.1
8.2
8.3
8.4
7.5
7.6
7.7
7.8
7.9
7.0
7.1
7.2
7.3
7.4
6.5
6.6
6.7
6.8
6.9
6.0
6.1
6.2
6.3
6.4
5.5
5.6
5.7
5.8
5.9
5.0
5.1
5.2
5.3
5.4
Ln for n = 1.0
0.978
0.983
0.987
0.992
0.996
0.955
0.960
0.964
0.969
0.974
0.930
0.935
0.940
0.945
0.950
0.904
0.909
0.914
0.920
0.925
0.876
0.881
0.887
0.893
0.898
0.846
0.852
0.858
0.864
0.870
0.814
0.820
0.827
0.833
0.839
0.779
0.786
0.793
0.800
0.807
0.741
0.749
0.757
0.764
0.772
0.700
0.708
0.717
0.725
0.733
1
0.979
0.983
0.988
0.992
0.997
0.955
0.960
0.965
0.969
0.974
0.930
0.936
0.941
0.945
0.950
0.904
0.910
0.915
0.920
0.925
0.876
0.882
0.888
0.893
0.899
0.846
0.852
0.859
0.865
0.870
0.814
0.821
0.827
0.834
0.840
0.780
0.787
0.794
0.801
0.808
0.742
0.750
0.757
0.765
0.772
0.700
0.709
0.718
0.726
0.734
2
to 9.99
3
0.979
0.984
0.988
0.993
0.997
0.956
0.960
0.965
0.970
0.975
0.931
0.936
0.941
0.946
0.951
0.905
0.910
0.915
0.921
0.926
0.877
0.883
0.888
0.894
0.899
0.847
0.853
0.859
0.865
0.871
0.815
0.822
0.828
0.834
0.841
0.780
0.787
0.794
0.801
0.808
0.743
0.751
0.758
0.766
0.773
0.702
0.711
0.719
0.727
0.735
4
0.980
0.984
0.989
0.993
0.997
0.956
0.961
0.966
0.970
0.975
0.931
0.937
0.942
0.946
0.951
0.905
0.911
0.916
0.921
0.926
0.877
0.883
0.889
0.894
0.900
0.848
0.854
0.860
0.866
0.872
0.816
0.822
0.829
0.835
0.841
0.781
0.788
0.795
0.802
0.809
0.744
0.751
0.759
0.766
0.774
0.702
0.710
0.719
0.728
0.736
5
0.980
0.985
0.989
0.993
0.998
0.957
0.961
0.966
0.971
0.975
0.932
0.937
0.942
0.947
0.952
0.906
0.911
0.916
0.922
0.927
0.878
0.884
0.889
0.895
0.900
0.848
0.854
0.860
0.866
0.872
0.816
0.823
0.829
0.836
0.842
0.782
0.789
0.796
0.803
0.810
0.744
0.752
0.760
0.767
0.775
0.703
0.712
0.720
0.728
0.736
6
0.980
0.985
0.989
0.994
0.998
0.957
0.962
0.967
0.971
0.976
0.932
0.938
0.943
0.947
0.952
0.906
0.912
0.917
0.922
0.927
0.879
0.884
0.890
0.895
0.901
0.849
0.855
0.861
0.867
0.873
0.817
0.823
0.830
0.836
0.843
0.782
0.790
0.797
0.803
0.810
0.745
0.753
0.760
0.768
0.775
0.704
0.713
0.721
0.729
0.737
7
0.981
0.985
0.990
0.994
0.999
0.958
0.962
0.967
0.972
0.976
0.933
0.938
0.943
0.948
0.953
0.907
0.912
0.918
0.923
0.928
0.879
0.885
0.890
0.896
0.901
0.849
0.856
0.861
0.867
0.873
0.818
0.824
0.831
0.837
0.843
0.783
0.790
0.797
0.804
0.811
0.746
0.754
0.761
0.769
0.776
0.705
0.713
0.722
0.730
0.738
8
0.981
0.986
0.990
0.995
0.999
0.958
0.963
0.968
0.972
0.977
0.933
0.939
0.943
0.948
0.953
0.907
0.913
0.918
0.923
0.928
0.880
0.885
0.891
0.897
0.902
0.850
0.856
0.862
0.868
0.874
0.818
0.825
0.831
0.838
0.844
0.784
0.791
0.798
0.805
0.812
0.747
0.754
0.762
0.769
0.777
0.706
0.714
0.723
0.731
0.739
9
0.982
0.986
0.991
0.995
1.000
0.959
0.963
0.968
0.973
0.977
0.934
0.939
0.944
0.949
0.954
0.908
0.913
0.919
0.924
0.929
0.880
0.886
0.892
0.897
0.903
0.851
0.857
0.863
0.869
0.874
0.819
0.825
0.832
0.838
0.844
0.785
0.792
0.799
0.806
0.812
0.747
0.755
0.763
0.770
0.777
0.707
0.715
0.723
0.732
0.740
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
4
8 12 16 20 24 28 32 36
9 18 27 36 45 54 63 72 81
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
4
8 12 16 20 24 28 32 36
9 18 27 36 45 54 63 72 81
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
4
8 12 16 20 24 28 32 36
9 18 27 36 45 54 63 72 81
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
×
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8 10 12 14 16 18
2
4
6
8 10 12 14 16 18
3
6
9 12 15 18 21 24 27
3
6
9 12 15 18 21 24 27
4
8 12 16 20 24 28 32 36
1
2
3
4
5
6
7
8
9
4
8 12 16 20 24 28 32 36
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
© Frank Tapson 2004 [trolFG:16]
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
5 10 15 20 25 30 35 40 45
6 12 18 24 30 36 42 48 54
7 14 21 28 35 42 49 56 63
8 16 24 32 40 48 56 64 72
9 18 27 36 45 54 63 72 81
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
216
288
360
432
504
576
648
213
284
355
426
497
568
639
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
219
292
365
438
511
584
657
64
96
128
160
192
224
256
288
32 × 2 =
66
99
132
165
198
231
264
297
33 × 2 =
=
=
=
=
=
=
=
3
4
5
6
7
8
9
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
158
237
316
395
474
553
632
711
154
231
308
385
462
539
616
693
×
×
×
×
×
×
×
=
=
=
=
=
=
=
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
33
33
33
33
33
33
33
=
=
=
=
=
=
=
3
4
5
6
7
8
9
77
77
77
77
77
77
77
2
3
4
5
6
7
8
9
×
×
×
×
×
×
×
32
32
32
32
32
32
32
225
300
375
450
525
600
675
=
=
=
=
=
=
=
75
75
75
75
75
75
75
222
296
370
444
518
592
666
79 ×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
152
228
304
380
456
532
608
684
79
79
79
79
79
79
79
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
156
234
312
390
468
546
624
702
162
243
324
405
486
567
648
729
×
×
×
×
×
×
×
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
31
31
31
31
31
31
31
76
76
76
76
76
76
76
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
3
4
5
6
7
8
9
62
93
124
155
186
217
248
279
31 × 2 =
78 ×
×
×
×
×
×
×
×
81 ×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
60
90
120
150
180
210
240
270
46
69
92
115
138
161
184
207
78
78
78
78
78
78
78
81
81
81
81
81
81
81
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
50
75
100
125
150
175
200
225
160
240
320
400
480
560
640
720
3
4
5
6
7
8
9
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
×
×
×
×
×
×
×
×
×
×
×
×
×
×
58
87
116
145
174
203
232
261
29
29
29
29
29
29
29
29 × 2 =
23
23
23
23
23
23
23
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
30
30
30
30
30
30
30
=
=
=
=
=
=
=
30 × 2 =
3
4
5
6
7
8
9
25 ×
×
×
×
×
×
×
×
80 ×
×
×
×
×
×
×
×
75 × 2 = 150
73
73
73
73
73
73
73
×
×
×
×
×
×
×
56
84
112
140
168
196
224
252
28
28
28
28
28
28
28
28 × 2 =
44
66
88
110
132
154
176
198
25
25
25
25
25
25
25
80
80
80
80
80
80
80
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
48
72
96
120
144
168
192
216
54
81
108
135
162
189
216
243
74
74
74
74
74
74
74
3
4
5
6
7
8
9
73 × 2 = 146
×
×
×
×
×
×
×
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
77 ×
×
×
×
×
×
×
×
22
22
22
22
22
22
22
2
3
4
5
6
7
8
9
27 ×
×
×
×
×
×
×
×
76 ×
×
×
×
×
×
×
×
24 ×
×
×
×
×
×
×
×
52
78
104
130
156
182
208
234
23 ×
×
×
×
×
×
×
×
24
24
24
24
24
24
24
=
=
=
=
=
=
=
=
22 ×
×
×
×
×
×
×
×
26 ×
×
×
×
×
×
×
×
74 × 2 = 148
72
72
72
72
72
72
72
72 × 2 = 144
71
71
71
71
71
71
71
98
147
196
245
294
343
392
441
49 × 2 =
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
150
200
250
300
350
400
450
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
153
204
255
306
357
408
459
51
51
51
51
51
51
51
49
49
49
49
49
49
49
=
=
=
=
=
=
=
51 × 2 = 102
=
=
=
=
=
=
=
3
4
5
6
7
8
9
50
50
50
50
50
50
50
3
4
5
6
7
8
9
×
×
×
×
×
×
×
94
141
188
235
282
329
376
423
47
47
47
47
47
47
47
47 × 2 =
50 × 2 = 100
×
×
×
×
×
×
×
=
=
=
=
=
=
=
96
144
192
240
288
336
384
432
48
48
48
48
48
48
48
3
4
5
6
7
8
9
48 × 2 =
×
×
×
×
×
×
×
92
138
184
230
276
322
368
414
46
46
46
46
46
46
46
46 × 2 =
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
156
208
260
312
364
416
468
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
162
216
270
324
378
432
486
56
56
56
56
56
56
56
×
×
×
×
×
×
×
3
4
5
6
7
8
9
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
159
212
265
318
371
424
477
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
165
220
275
330
385
440
495
57
57
57
57
57
57
57
168
224
280
336
392
448
504
×
×
×
×
×
×
×
171
228
285
342
399
456
513
=
=
=
=
=
=
=
3
4
5
6
7
8
9
57 × 2 = 114
55
55
55
55
55
55
55
55 × 2 = 110
53
53
53
53
53
53
53
53 × 2 = 106
=
=
=
=
=
=
=
56 × 2 = 112
54
54
54
54
54
54
54
54 × 2 = 108
52
52
52
52
52
52
52
52 × 2 = 104
190
285
380
475
570
665
760
855
2
3
4
5
6
7
8
9
71 × 2 = 142
=
=
=
=
=
=
=
=
27
27
27
27
27
27
27
210
280
350
420
490
560
630
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
95 ×
×
×
×
×
×
×
×
95
95
95
95
95
95
95
26
26
26
26
26
26
26
3
4
5
6
7
8
9
188
282
376
470
564
658
752
846
194
291
388
485
582
679
776
873
198
297
396
495
594
693
792
891
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
70
70
70
70
70
70
70
2
3
4
5
6
7
8
9
97 ×
×
×
×
×
×
×
×
97
97
97
97
97
97
97
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
192
288
384
480
576
672
768
864
99
99
99
99
99
99
99
99 ×
×
×
×
×
×
×
×
196
294
392
490
588
686
784
882
70 × 2 = 140
94 ×
×
×
×
×
×
×
×
94
94
94
94
94
94
94
96 ×
×
×
×
×
×
×
×
96
96
96
96
96
96
96
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
×
1
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
98 ×
×
×
×
×
×
×
×
98
98
98
98
98
98
98
© Frank Tapson 2004 [trolFG:17]
Multiplication Tables
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
2
4
6
8
10
12
14
16
18
3
6
9
12
15
18
21
24
27
4
8
12
16
20
24
28
32
36
5
10
15
20
25
30
35
40
45
6
12
18
24
30
36
42
48
54
7
14
21
28
35
42
49
56
63
8
16
24
32
40
48
56
64
72
9
18
27
36
45
54
63
72
81
=
=
=
=
=
=
=
246
328
410
492
574
656
738
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
252
336
420
504
588
672
756
=
=
=
=
=
=
=
249
332
415
498
581
664
747
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
255
340
425
510
595
680
765
40
60
80
100
120
140
160
180
20 × 2 =
42
63
84
105
126
147
168
189
21 × 2 =
270
360
450
540
630
720
810
3
4
5
6
7
8
9
=
=
=
=
=
=
=
273
364
455
546
637
728
819
279
372
465
558
651
744
837
=
=
=
=
=
=
=
3
4
5
6
7
8
9
258
344
430
516
602
688
774
67 ×
×
×
×
×
×
×
×
65
65
65
65
65
65
65
=
=
=
=
=
=
=
2
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
134
201
268
335
402
469
536
603
130
195
260
325
390
455
520
585
×
×
×
×
×
×
×
93
93
93
93
93
93
93
276
368
460
552
644
736
828
=
=
=
=
=
=
=
3
4
5
6
7
8
9
92
92
92
92
92
92
92
45
60
75
90
105
120
135
15
15
15
15
15
15
15
42
56
70
84
98
112
126
=
=
=
=
=
=
=
3
4
5
6
7
8
9
×
×
×
×
×
×
×
20
20
20
20
20
20
20
261
348
435
522
609
696
783
=
=
=
=
=
=
=
3
4
5
6
7
8
9
87
87
87
87
87
87
87
128
192
256
320
384
448
512
576
67
67
67
67
67
67
67
93 × 2 = 186
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
132
198
264
330
396
462
528
594
×
×
×
×
×
×
×
91
91
91
91
91
91
91
91 × 2 = 182
267
356
445
534
623
712
801
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
138
207
276
345
414
483
552
621
92 × 2 = 184
=
=
=
=
=
=
=
=
=
=
=
=
=
=
64
64
64
64
64
64
64
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
3
4
5
6
7
8
9
3
4
5
6
7
8
9
66 ×
×
×
×
×
×
×
×
2
3
4
5
6
7
8
9
3
4
5
6
7
8
9
×
×
×
×
×
×
×
×
×
×
×
×
×
×
70
105
140
175
210
245
280
315
66
66
66
66
66
66
66
69 ×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
90
90
90
90
90
90
90
90 × 2 = 180
89
89
89
89
89
89
89
89 × 2 = 178
=
=
=
=
=
=
=
=
74
111
148
185
222
259
296
333
69
69
69
69
69
69
69
15 × 2 = 30
39
52
65
78
91
104
117
264
352
440
528
616
704
792
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
136
204
272
340
408
476
544
612
14
14
14
14
14
14
14
=
=
=
=
=
=
=
=
=
=
=
=
=
=
35
35
35
35
35
35
35
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
14 × 2 = 28
3
4
5
6
7
8
9
3
4
5
6
7
8
9
37 ×
×
×
×
×
×
×
×
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
×
×
×
×
×
×
×
×
×
×
×
×
×
×
68
102
136
170
204
238
272
306
37
37
37
37
37
37
37
68 ×
×
×
×
×
×
×
×
3
4
5
6
7
8
9
13
13
13
13
13
13
13
13 × 2 = 26
88
88
88
88
88
88
88
88 × 2 = 176
=
=
=
=
=
=
=
=
72
108
144
180
216
252
288
324
68
68
68
68
68
68
68
×
×
×
×
×
×
×
= 36
= 48
= 60
= 72
= 84
= 96
= 108
33
44
55
66
77
88
99
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
78
117
156
195
234
273
312
351
21
21
21
21
21
21
21
3
4
5
6
7
8
9
=
=
=
=
=
=
=
34
34
34
34
34
34
34
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
×
×
×
×
×
×
×
3
4
5
6
7
8
9
36 ×
×
×
×
×
×
×
×
39 ×
×
×
×
×
×
×
×
3
4
5
6
7
8
9
12
12
12
12
12
12
12
12 × 2 = 24
×
×
×
×
×
×
×
82
123
164
205
246
287
328
369
36
36
36
36
36
36
36
76
114
152
190
228
266
304
342
=
=
=
=
=
=
=
11
11
11
11
11
11
11
11 × 2 = 22
=
=
=
=
=
=
=
=
86
129
172
215
258
301
344
387
=
=
=
=
=
=
=
=
3
4
5
6
7
8
9
30
40
50
60
70
80
90
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
38 ×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
41
41
41
41
41
41
41
2
3
4
5
6
7
8
9
90
135
180
225
270
315
360
405
19
19
19
19
19
19
19
3
4
5
6
7
8
9
43 ×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
×
×
×
×
×
×
×
38
57
76
95
114
133
152
171
19 × 2 =
×
×
×
×
×
×
×
80
120
160
200
240
280
320
360
43
43
43
43
43
43
43
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
10
10
10
10
10
10
10
10 × 2 = 20
=
=
=
=
=
=
=
=
84
126
168
210
252
294
336
378
45 ×
×
×
×
×
×
×
×
3
4
5
6
7
8
9
=
=
=
=
=
=
=
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
45
45
45
45
45
45
45
×
×
×
×
×
×
×
3
4
5
6
7
8
9
40
40
40
40
40
40
40
2
3
4
5
6
7
8
9
88
132
176
220
264
308
352
396
18
18
18
18
18
18
18
×
×
×
×
×
×
×
34
51
68
85
102
119
136
153
17
17
17
17
17
17
17
17 × 2 =
42 ×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
×
×
×
×
×
×
×
= 48
= 64
= 80
= 96
= 112
= 128
= 144
36
54
72
90
108
126
144
162
3
4
5
6
7
8
9
18 × 2 =
×
×
×
×
×
×
×
118
177
236
295
354
413
472
531
42
42
42
42
42
42
42
2
3
4
5
6
7
8
9
87 × 2 = 174
85
85
85
85
85
85
85
16
16
16
16
16
16
16
16 × 2 = 32
=
=
=
=
=
=
=
=
122
183
244
305
366
427
488
549
44 ×
×
×
×
×
×
×
×
×
×
×
×
×
×
×
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
44
44
44
44
44
44
44
86
86
86
86
86
86
86
×
×
×
×
×
×
×
85 × 2 = 170
83
83
83
83
83
83
83
83 × 2 = 166
59
59
59
59
59
59
59
2
3
4
5
6
7
8
9
126
189
252
315
378
441
504
567
65 ×
×
×
×
×
×
×
×
61 ×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
64 ×
×
×
×
×
×
×
×
116
174
232
290
348
406
464
522
61
61
61
61
61
61
61
2
3
4
5
6
7
8
9
35 ×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
120
180
240
300
360
420
480
540
63 ×
×
×
×
×
×
×
×
34 ×
×
×
×
×
×
×
×
2
3
4
5
6
7
8
9
=
=
=
=
=
=
=
=
63
63
63
63
63
63
63
41 ×
×
×
×
×
×
×
×
58
58
58
58
58
58
58
2
3
4
5
6
7
8
9
124
186
248
310
372
434
496
558
40 ×
×
×
×
×
×
×
×
60 ×
×
×
×
×
×
×
×
=
=
=
=
=
=
=
=
59 ×
×
×
×
×
×
×
×
60
60
60
60
60
60
60
2
3
4
5
6
7
8
9
58 ×
×
×
×
×
×
×
×
62 ×
×
×
×
×
×
×
×
86 × 2 = 172
84
84
84
84
84
84
84
84 × 2 = 168
3
4
5
6
7
8
9
2
3
4
5
6
7
8
9
×
×
×
×
×
×
×
39
39
39
39
39
39
39
82
82
82
82
82
82
82
2
3
4
5
6
7
8
9
82 × 2 = 164
38
38
38
38
38
38
38
62
62
62
62
62
62
62
© Frank Tapson 2004 [trolFG:18]
An alternative front cover for 'A Book of Multiplication Tables 10 to 99
Cut out and paste over the existing front cover, before making multiple copies.
A Book of
Multiplication
Tables
10 to 99
© Frank Tapson 2004 [trolFG:19]