MATHEMATICAL RECREATIONS 185

Write these two series of numbers, omitting ¢,;1, column-
wise and add a third column consisting of the number b and
the numbers obtaind from it by successiv doubling, as repre-
sented below :

a ( == {Zo) Qo b
q1 A 2b
g2 Qs 22p
qs as 23b
q'n @ 2n)

In case @ = 1, ¢4 = 0, whence » = 0, and we will have only
one number in each column.

Since the numbers ao, a1, @2, ** -, a, are all positiv or zero
(§ 78) and less than 7, which is 2 (§ 80), these numbers are all
either 0 or 1. Moreover, if any number in the first column
is even, the division is exact and the corresponding remainder,
which is the number in the same row in the second column,
is 0. If any number in the first column is odd, the division
is inexact and the corresponding remainder, that is, the corre-
sponding number in the second column, is 1. Moreover, since
a, = ¢»—= 0 (§ 101), both @, and g, are equal to 1.

Now we have by § 101

¢ =a+ a1°2 + a2022 4+ a3-23 4 -+ 4 a,-2".

Here we have a changed into the binary scale.

Then ab = aeb + a1(2b) + a2(220) + a3(23%) + - -+ + a.(27).

Each of the numbers ao, a1, @, @3, *++, @, is either 0 or 1.

If any one of them is 0, the corresponding term can be crossed
out. If any one of them is 1, it may be omitted as a factor
from the corresponding term.

Thus ab-is the sum of those terms in our last column,

b, 20, 2%h 2% wua s 2P0

which correspond to a value 1 in the second column, or to odd
numbers in the first column.
Our rule is therfor proved.