Definition:Set of Residue Classes/Least Absolute

Definition

Let $\eqclass a m$ be the residue class of $a$ (modulo $m$).

Except when $r = \dfrac m 2$, we can choose $r$ to be the integer in $\eqclass a m$ which has the smallest absolute value.

In that exceptional case we have:

$-\dfrac m 2 + m = \dfrac m 2$

and so:

$-\dfrac m 2 \equiv \dfrac m 2 \pmod m$

Thus $r$ is defined as the least absolute residue of $a$ (modulo $m$) if and only if:

$-\dfrac m 2 < r \le \dfrac m 2$

Also see

• Definition:Least Positive Residue