MATHEMATICAL RECREATIONS 195

Thus 1=a,—ar=r—1
Hence 0=(@n—a) —1=r—2
That is, 0=s=r—2
Since r—1=r—1=r-—-1
and r—2=s=0,
1=@—-1)—-—5s=r-—1
That is, 1=Z2t=r—-1

Since0=s=7r—2and 1 =¢t=r — 1, s and ¢ are digits.
Since

a—a =t+ Y (r — Drk1 4 srm,
k=2

the highest and lowest digits of @ — a’ are respectivly s and ¢
and all the other digits, if there are any, are equal to r — 1.

481. For the decimal system the above theorem becomes
the following:

Th. In the decimal system if we interchange the first and last
digits of a number of m digits (m > 1), of which the first is
greater than the last, and subtract 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 8, but no larger;
the lowest digit will be the difference between 10 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 9; the sum of the highest and lowest digits will be 9; and all
the other digits, if there are any, that is, if m > 2, will be 9’s.

Examples: 52 561 150 46322
=iy 165 051 26324
i 396 099 19998

482. Now consider any number d of m digits (m > 1), of
which the first may be 0 and such that the sum of the first
and last digits, s and ¢, is » — 1, and all the other digits, if
there are any, are equal to r — 1.

As before, let m = n + 1, where n = 1