186 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

It may be noticed that this proof does not depend on the
numbers to be multiplied being written in any special radix
notation. It is as good for numbers written in the septimal
or in the sexidenal notation as for numbers written in the
decimal system. However in applying the rule, if the radix
is odd, one must be careful in picking out the odd numbers
in the first column. With an odd radix a number is not neces-
sarily odd if its units digit is odd ; and it may be odd if its
units digit is even. For example, in the septimal scale 23 is
odd but 33 is even; and 32 is odd.

Ex. 3. Multiply 96 and 87 by the Russian peasant method.
Check by multiplying 87 and 96 by the same method.

Ex. 4. Multiply 783 and 436 by this method. Check by
congruences to the modulus 5. Check also by the moduli
4 and 6.

Ex. 5. Multiply 55(8) and 30(8) by the Russian peasant
method. Check by changing the given numbers and product
into the decimal system.

Ex. 6. Multiply 63(7) and 33(7) by this method. Check

as in ex. S.

471. Some of my readers may be interested in the following
two rules, which are variations on the Russian peasant rule.

472. Rule 2. The Generalized Binary Rule for Multipli-
cation. 7o multiply a by b write down in a vertical column the
number a and a series of successiv quotients, exact, lower, or
upper, obtaind by dividing it and the series of quotients by 2,
until you obtain the quotient 0.

In another column, starting on a line with the first quotient,
write down the number b and the numbers obtaind from it by
successiv doubling, until you have a number in the second column
for each quotient in the first.

Next add the numbers in the second column which correspond
to lower quotients and those which correspond to upper quotients.
Subtract the latter sum from the former.

The result is a X b.