192 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

478. This rule is a particular instance of a general rule,
obtaind from the above by putting ““m (two or more)” for
‘“three,” the result being for a number of

2 digits 9 X 11 =9 X 11 X 1 =99 X 1= 99
g e x-Sl =0 X 11X 11=9X "il= 1080
4 “ OX 1221 =9 X 11 X 111 =99 X 111 = 10989
5 “ 0X12221 =9 X 11 X 1111 = 99 X 1111 = 109989

For a number of m digits (m > 1) the result is 9 times a
number of m digits, of which the first and last are 1's and the
others, if there are any, are 2’s; or 99 times a number of m — 1
digits, all of which are I's; or, if m = 2, the result is 99, if
m > 2, the result is a number of m + 1 digits, of which the first
two are 1, 0, the last two 8, 9, and the others, if there are any,
are all 9's. Also, if m = 3, we note that the result is 32 X 112,
or 332

479. The above rule is for numbers written in the decimal
system. We will prove the corresponding rule for numbers
written in any radix notation, which will include this rule as -
a particular case.

First we will prove the following theorem:

480. Th. If a number is written in a radix notation whose
base is r and has m digits (m > 1), the first of which is greater
than the last, if we interchange the first and last digits and sub-
tract the resulting number from the given number, then the highest
digit of the difference, considerd also as an m-digit number, will
be one less than the difference of the highest and lowest digits of
the given number, a number that may be as small as 0 and may
be as large as r — 2, but no larger; the lowest digit will be the
difference between the radix and the difference between the highest
and lowest digits of the given number, a number that may be as
small as 1, but no smaller, and may be as large as r — 1; the sum
of the highest and lowest digits will be r — 1; and all the other
digits, if there are any, that is, if m > 2, will be equal tor — 1.

To prove this theorem, let @ be any number of m digits
(m > 1), the first digit being greater than the last.