CONGRUENCE 85

LEAST COMMON MULTIPLE

172. Def. a X b stands for the least positiv common mul-
tiple of @ and b and is read “@ mul b.”

For example, 2X3=06,- 4X6=12 9X3=29,
(— 12) X 18 = 36.

There is no least common positiv, zero, or negativ multiple
of two integers; for example,

vee, 1812, 6,0, — 6, =12, — A8, «4

are all common multiples of 2 and 3. But there is no least
of these. On this account the word ‘‘positiv”’ is usually
omitted from the phrase ‘least positiv common multiple,” but
is always understood.

173. Th. a—-=0~b—-—=0Da Xb=0bXa

174. Th. a—-=0 ~a > b D a X b = |al.
Example. (— 6) X 2 = 6.

178. Th. a—=0>Da X 1 = {al.
Example. 7 X 1 =17.

CONGRUENCE

176. Def. When we say of three integers a, b, and m that
““a is congruent to b with respect to the modulus 7"’ we mean
that a — b > m.

The idea of “being congruent to with respect to a modu-
lus” is a relation that may exist among three numbers, a
triadic relation. The statement that this relation exists among
a, b, and m, that is, that ‘‘@ is congruent to b with respect to
the modulus m”’ we will write ¢ = b (mod m). This state-
ment we will call a congruence, a the left member, b the right,
and m the modulus of the congruence.

Thus [ea =b (modm)] =[a—0>m];
that is, if a = b (mod m), thena — 0 > m
and, if a — b>> m, then @ = b (mod m) (1).7

(1) The idea of congruence is due to Karl Friedrich Gauss. See his
Disquisitiones Arithmeticae, p. 1.