Definition:Set of Residue Classes/Least Positive

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

Let $r$ be the smallest non-negative integer in $\eqclass a m$.

Then from Integer is Congruent to Integer less than Modulus:
 * $0 \le r < m$

and:
 * $a \equiv r \pmod m$

Then $r$ is called the least positive residue of $a \pmod m$.

Also known as
Some sources call this the common residue.

Others call it the least non-negative residue.

Some sources use the term the residue, and do classify other elements of $\eqclass a m$ as residues.

Also see

 * Definition:Least Absolute Residue