184 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

lower quotient obtaind by dividing it by 2, and so on, until you
obtain the quotient 1.

Under b write its double, under this double its double, and so
on, until you have as many numbers in the second column as in
the first.

Next add the numbers in the second column which correspond
to odd numbers in the first.

The result is the product of a and b.

The rule may be illustrated by the following examples:

r- ks A5 24 Ex. 2. 42 X b—Omit
22 48—Omit 21 2b
11 926 10 4b—Omit
5 192 5 8b
2 384—0Omit 2 166—O0Omit
1 768 1 32b
1080 42b

The proof depends on changing 42 into the binary scale by
the method of § 102. If the a of that theoremis42and » = 2,
on successiv division as described in that article we have the
six quotients 21, 10, 5, 2, 1, 0 and remainders 0, 1, 0, 1, 0, 1.

Hence 42=0+1:2+40-2241-2340-2¢4 4 1.25

=2+ 23 4 25

Therfor 420 = (2 + 23 4 2%b = 2b + 8b + 32b

This example proves the Russian peasant rule for any value
of b when a = 42.

To prove the rule for all values of ¢ and b, let » = 2 and
divide a and the successiv quotients as explaind in § 102.
We thus get two series of numbers, first the series

a, q1, G2, 93, ***s Qs ***» Iny Gnt1, Where n = 0,

consisting of a and the successiv quotients, only the last quo-
tient, ¢,.1, being zero, and second the series of remainders

aOy ai, Qs, L =3 Ap=3y ~22;Cn;

the remainder a;_; corresponding to the quotient g;.