# Definition:Congruence (Number Theory)/Residue

## Definition

Let $m \in \Z_{\ne 0}$ be a non-zero integer.

Let $a, b \in \Z$.

Let $a \equiv b \pmod m$.

Then $b$ is a residue of $a$ modulo $m$.

Residue is another word meaning remainder, and is any integer congruent to $a$ modulo $m$.

## Also defined as

Some sources define the residue to be the smallest (non-negative) integer congruent to $a$ modulo $z$, that is, what on $\mathsf{Pr} \infty \mathsf{fWiki}$ is designated as the least positive residue.