Definition:Congruence (Number Theory)/Integers/Remainder after Division

Definition
Let $m \in \Z_{> 0}$ be an integer. Congruence modulo $m$ is defined as the relation $\equiv \pmod m$ on the set of all $a, b \in \Z$:
 * $a \equiv b \pmod m := \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$

That is, such that $a$ and $b$ have the same remainder when divided by $m$.

Also see

 * Equivalence of Definitions of Congruence


 * Congruence Modulo $m$ is Equivalence Relation