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

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 6$: Examples of Finite Groups: $\text{(iii)}$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 2.5$. Congruence of integers