86 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

For example, 8 = 4 (mod 2) for 8 —4>2
12 = 2 (mod 5) for12 — 2> 5
6 = 18 (mod 12) for — 12> 12

7 =7 (mod 3) for RS

—= 6 (mod 3) for 1->33

177. Th. From the definition of congruence it follows that
the modulus cannot be zero, for no number is divisible by zero.

178. Frequently we discuss congruences which have the
same modulus. In such case we abbreviate the statement
a = b (mod m) to @ = b, the modulus being understood. The
relation of congruence is in this case a relation between two
numbers, a dyadic relation.

From the definition of congruence we can easily prove the
following theorems.

179. Th. [a = 0 (mod m)] = [a > m]
For example, since 9 > 3, 9 = 0 (mod 3)

180. Th. a=b(modm) Da =b~— |a —b| -< |m|
Fora =b(modm) Da —b>3>m
' Da—b=0~—|la—0b|-< |m| §72.

181. Th. a—-=b~ |ea —b| < |m| D a-= b (mod m)
For example, 15 —= 17 (mod 6).

182. Th. a = b (mod &= I). That is, every integer is con-
gruent to every integer with respect to plus or minus unity as a
modulus.

Fora—b>1anda — 06> — 1.

On this account the modulus is generally assumed to be
neither 1 nor — 1.

183. Th. a =a (mod m). Or, every integer is congruent
to itself with respect to any modulus.
Fora —a=0>3>m>Da—a>m.

184. Th. Ifa = b, then a = b with respect to any modulus.
Fora=b2Da—0=02m