Definition:Congruence (Number Theory)/Integers/Integer Multiple

Definition
Let $m \in \Z_{> 0}$ be an integer. Let $x, y \in \Z$.

$x$ is congruent to $y$ modulo $m$ their difference is an integer multiple of $m$:
 * $x \equiv y \pmod m \iff \exists k \in \Z: x - y = k m$

In terms of divisibility, this can be rendered:


 * $x \equiv y \pmod m \iff m \divides \paren {x - y}$

Also see

 * Equivalence of Definitions of Congruence


 * Congruence Modulo $m$ is Equivalence Relation