146 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

transposing the last of the m digits over the others, replacing it
by its opposit, and annexing a zero. § 357.
Thus 472000000 = 72400000 = 2470000 = 472000 = 72400
= 2470 = 472 (mod 73 4 1)

361. Th. If the modulus is r™ + 1, any number, all of whose
digits except m consecutiv digits are zeros, is congruent to the
number obtaind by replacing the first q of the m digits by their
opposits, interchanging these opposits with the last m — q, keeping
the cyclic order of the digits unchanged, and omitting q zeros or
annexing 2m — q zeros. § 332.

362. Th. If the modulus is r™ 4 1, any number, all of whose
digits except m consecutiv digits are zeros, is congruent to the
number obtaind by replacing the first q of the m digits by their
opposits, interchanging these opposits with the lastm — q, keeping
the cyclic order unchanged, and omitting a set of zeros whose
number is congruent to q (mod 2m) or annexing a set of zeros
whose number is congruent to 2m — q (mod 2m).

The proof is like that of § 339, using 2m and »™ 4+ 1 for
moduli instead of m and 7 — 1 respectivly. § 300.

363. As a consequence of § 358 or § 359 we have the follow-
ing theorem.

Th. If the modulus is r™ + 1, any number, all of whose
m-figure periods except one consist of zeros, is congruent to the
number obtaind by changing the sign of that period and moving
it an odd number of periods to the right or left, or by moving it
an even number of periods to the right or left without change
of sign.

For example,

27,00,00,00,00 = — 27,00,00,00 = 27,00,00 = — 27,00 = 27

(mod 72 + 1)

365,000 = — 365 (mod 7® + 1)

364. Th. If po, p1, po, P3, =+ are the successiv m-figure
periods of a number, po being the period containing the units
digit, then

a = Py — (p1 — (p2a — (ps — +>*))) (mod r + 1)