140 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

340. As a particular case of either § 329 or § 339 we have
the following 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 moving this period any number of periods to
the right or left.

For example, 36,00,00 = 36,00 = 36 (mod 72 — 1)

341. Th. If po, p1, P2y Ps, **+, Pn are the successiv m-figure
periods of a number a, po being the period containing the units
digit, then

a=po+p1r+ p2+ ps+ -0 + pn (mod rm — 1)
or a = i (pr) (mod r™ — 1)
k=0

This can be proved from §336 or from §313,
since ™ = 1 (mod ™ — 1)

Otherwise stated: If the modulus is ™ — 1, any number is
congruent to the sum of its m-figure periods.

For example, 23157=24+34+1+454+7(modr — 1)
32,13,20,23 = 32 + 13 + 20 + 23 (mod 72 — 1)
271,103,512 =271 4+ 103+ 512 (mod 7* — 1)

m—1
342. Th. o = > [( 3 ak) r’] (mod r™ — 1)
k=l (mod m)

=0

Def. Here the symbol > a;, stands for the sum of all
k=l (mod m)

the a;’s whose subscripts are congruent to /, modulus .

For example, taking m = 6,
213233,412365 = 625598 (mod 7* — 1)

This theorem is a consequence of the preceding theorem.

343. Th. By the process of § 341 any number whose digits
are all positiv or zero, or all negativ or zero, and which has more
than one m-figure period may be reduced to a one-period number
with respect to the modulus r™ — 1.