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 for 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.

Also see

• Definition:Least Positive Residue
• Definition:Complete Residue System
• Definition:Residue Class

Sources

• 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-2}$ Residue Systems: Definition $\text {4-2}$
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): congruence