CONGRUENCE 145

357. Th. If the modulus is r™ + 1, any number may be
subtracted from any digit, if the same number is subtracted from
or added to another digit, provided that and according as the
number of places from one digit to the other is an odd or an even
multiple of m.

The proof is like that of § 334, using the fact that
rm = — 1 (mod ™ + 1). Apply § 309.

Ex. 1. 365 145 (mod 7 + 1)

Ex. 2 365 167 (mod » 4+ 1)

Ex. 3 47357 = 27157 (mod 72 + 1)

Ex. 4. 47357 = 27359 (mod 7% 4 1)
5
6

i1

Ex. 4566546 = 1563546 (mod »* 4 1)
Ex. 4566546 = 1566549 (mod 7* + 1)

358. Th. If the modulus is r™ + 1, any digit may be replaced
by 0 and subtracted from or added to any other digit, such that
and according as the number of places from one digit to the other
is an odd or an even multiple of m.

By T 364 = 34 (mod7r 4+ 1)

Ex. 2 364 = 67 (mod 7 + 1)

Ex. 3 24765 = 4745 (mod r + 1)
Ex. 4. 24765 = 4767 (mod r + 1)
Ex. 5. 35724 = 33704 (mod 72+ 1)
Ex. 6 35724 = 5727 (mod 72 + 1)
Ex. 7 5273494 = 5203424 (mod 7* + 1)
Ex. 8 5273494 = 273499 (mod 7* + 1)

359. Th. If the modulus is r™ + 1, any of the m-figure
periods of a number may be replaced by a period of zeros and
subtracted from any period that is an odd number of periods away
or added to any period that is an even number of periods away.

For example, 25,43,79 = 25,00,36 (mod 72 4+ 1)

273,143,298,325 = 273,000,298,468 (mod 73 4 1)

360. Th. If the modulus is ™ + 1, any number, all of whose
digits except m consecutiv digits are zeros, is congruent to the
number obtaind by transposing the first of the m digits over the
others, replacing it by its opposit, and omitting a zero; or by