138 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Nowa =au+ - +ar*+ - +ar'+ -+ +a

Hence
a=a,r~+ - +ar*+ - +ap'+ -+ +a(modrm — 1)
Subtracting from this congruence the congruence

0 = br* — br! (mod ™ — 1),

we get
g=artt e Fl@m=0rr+ - +(@+br+4 .- +.a
For example, 576 = 396 (mod r — 1)

576 = 279 (mod » — 1)
27314 = 7514 (mod 7? — 1)
4723145 = 723149 (mod 73 — 1)

I

335. Th. If the modulus is r™ — 1, any digit may be replaced
by 0 and added to any other digit, provided the number of places
from one digit to the other is a multiple of m.

This follows from the preceding theorem by subtracting
from a digit the digit itself.

For example, 235 =55 (mod 7 — 1)

673235 = 73295 (mod 72> — 1)

336. Th. Ifthemodulusisr™ — 1,any of the m-figure periods
of a number may be replaced by a period of zeros and added to
any other period.

For example, 37,2541 = 25,78 (mod 72 — 1)

134,251 = 358,000 (mod 73 — 1)

337. Th. If the modulus is r™ — 1, any number, all of whose
digits except m consecutiv digits are zeros (and of which some
of the m digits may be zero), is congruent to the number obtaind
by transposing the first of the m digits over the others and omitting
a zero or by transposing the last of the m digits over the others
and annexing a zero.

Thus 354000000 = 54300000 = 4350000 = 354000 (mod * — 1)

This follows immediately from § 334.