136 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

328. The reader may be interested to interpret §§ 315-323
for the radices 2, 4, 8, 6, ©.

329. Th. If nis a positiv or zero multiple of m,

10"a = a (mod r™ — 1)
For 10" = =1 (mod ™ — 1)
Hence 10" =1 § 303.
Therfor 10"a = a § 212.

For example, 24,75,00,00,00 = 24,75 (mod 72 — 1)
10,263,000,000 = 10,263 (mod 73 — 1)

In words: If the modulus is r™ — 1, any number is congruent
to the number obtaind from it by omitting or annexing any number
of m-figure periods, each consisting of zeros.

As particular cases of this theorem we have the following
two theorems.

330. Th. If nis a positiv or zero integer,
10ma = a (modr — 1)

In words: If the modulus is the immediate predecessor of the
radix, any number is congruent to the number obtaind by omitting
or annexing any number of zeros.

For example,
37649201000 = 3764920100 = 37649201 (mod » — 1)
37604000 = 3760400 = 37604 (mod » — 1)

331. Th. If the modulus is the immediate predecessor of the
square of the radix, any number is congruent to the number
obtaind by omitting or annexing an even number of zeros.

332. Th. If nis a positiv or zero even multiple of m,
10"a = a (mod r™ + 1),
if m is a positiv odd multiple of m,

10" = — a (mod r™ + 1).