CONGRUENCE 137

This follows readily from § 308, since 10" = — 1 (mod 7™ 4 1)

For example,
15,327,000,000,000 = 15,327,000 = — 15,327 (mod 73 4 1)

In words: If the modulus is r™ + 1, any number is congruent
to the number obtaind by either omitting or annexing an even
number of m-figure periods consisting of zeros; and is congruent
to the number obtaind by changing its sign and either omitting
or annexing an odd number of m-figure periods consisting of
2eros.

As a particular case of this theorem we have the following
theorem.

333. Th. If n is a positiv or zero even number,
10Ma = a (modr + 1);
if n is a positiv odd number,
10"a = — a (mod r + 1)

In words: If the modulus is the immediate successor of the
radix, any number is congruent to the number obtaind by either
omatting or annexing an even number of zeros; and is congruent
to the number obtaind by changing its sign and either omitting
or annexing an odd number of zeros.

334. Th. If the modulus is r™ — 1, any number may be
subtracted from any digit, if the same number is added to another
digit, provided the number of places from one digit to the other
is a multiple of m.

Let a; and a; be any two digits such that the number of
places from one to the other is a multiple of .

Then k2 — 1> mand k =1 (mod m)
Hence, since rm =1 (mod r™ — 1),

r* = ¢! (mod r™ — 1) § 304.
Then br* = br! (mod r™ — 1) § 213.

And br¥ — br! = 0 (mod r™ — 1)