CONGRUENCE 139

338. 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 transposing the first q of the m digits over the
others (which is equivalent to transposing the last m — q over the
others or interchanging the first q with the last m — q), keeping
the cyclic order of the digits unchanged, and omitting q zeros or
annexing m — ¢ 2eros.

This follows immediately from § 337.

For example, 539780000 = 9785300 = 978530000000

(mod 7* — 1)

339. 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 transposing the first q of the m digits over the
others, keeping the cyclic order unchanged, and omitting a set of
seros whose number is congruent to q (mod m) or annexing a
set of zeros whose number is congruent to m — ¢ (mod m).

Let a be the given number, b the number represented by
the m digits, n the number of zeros following the m digits,
b’ the number obtaind from b by transposing the ¢ digits, y the
number of zeros omitted.

Then @ = b-10" and - 10"—¢ is the number obtaind from a
by transposing the first g of the m digits and omitting g zeros.

Hence a="b-10"1 (mod r» — 1) § 338.

But by hypothesis vy = ¢ (mod m)

Hence n —1vy =mn — ¢ (mod m)

Then, since 10™ = 1 (mod ™ — 1),

b'-10"v = b’-10" (mod r™ — 1) §§ 304, 213.

Therfor a=0b-10"¥ (mod r™ — 1)

Again let z be the number of zeros annext.

Then by hypothesis z = m — ¢ = — ¢ (mod m)

Hence n + 2z =n — ¢q (mod m) § 205.
Therfor b 10m+z = b’-102 (mod r™ — 1) §§ 304, 213.

I

For example,
(203)0000 = (320)00 = (320)00000000 (mod 7* — 1)
(045)000 = (504)0 = (504)0000 (mod 7* — 1)