134 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

314. Th. If the modulus is r™, where m is positiv, any num-
ber is congruent to its lowest m-figure period.

For l 1
= kZO (per*™) = po + kZl (Px(rm)*)

And 7 = 0 (mod »™)
Hence,if k=1, ) =0
Therfor a = po

315. Th. A number is congruent to its lowest m-figure period
with respect to any factor of the mth power of the radix as a
modulus.

316. Th. A number is divisible by any factor of the mith
power of the radix when, and only when, its lowest m-figure
period is so divisible.

317. Th. A number written as ordinarily with positiv or
zero digits all less than the radix is divisible by the mth power
of the radix when, and only when, its lowest m-figure period
consists of zeros.

For in this case the lowest m-figure period is less than the
mth power of the radix and no integer except zero is divisible
by an integer numerically greater than itself.

318. As special cases of these theorems we have the fol-
iowing:

Th. A number is congruent to its units digit with respect to
any factor of the radix as a modulus.

319. Th. A number is divisible by any factor of the radix
when, and only when, its units digit is so divisible.

320. Th. A number written with positiv or zero digits all less
than the radix is divisible by the radix when, and only when,
its units digit 1s zero.

321. Th. A number is congruent to its lowest two-figure period
with respect to any factor of the square of the radix as a modulus.