Definition:Congruence (Number Theory)/Integers

Definition
Let $m \in \Z_{> 0}$.

Then we define congruence modulo $m$ as the relation $\mathcal R_m$ on the set of all $a, b \in \Z$:
 * $\mathcal R_m = \left\{{\left({a, b}\right) \in \Z \times \Z: \exists k \in \Z: a = b + km}\right\}$

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

Definition by Integral Multiple
We also see that $a$ is congruent to $b$ modulo $m$ if their difference is a multiple of $m$:

Also see

 * Equivalence of Congruence Definitions


 * Congruence Modulo $m$ is an Equivalence Relation