# Definition:Residue Class

## Definition

Let $m \in \Z_{>0}$ be a (strictly) positive integer.

Let $\RR_m$ be the congruence relation modulo $m$ on the set of all $a, b \in \Z$:

$\RR_m = \set {\tuple {a, b} \in \Z \times \Z: \exists k \in \Z: a = b + k m}$

We have that congruence modulo $m$ is an equivalence relation.

So for any $m \in \Z$, we denote the equivalence class of any $a \in \Z$ by $\eqclass a m$, such that:

 $\ds \eqclass a m$ $=$ $\ds \set {x \in \Z: a \equiv x \pmod m}$ $\ds$ $=$ $\ds \set {x \in \Z: \exists k \in \Z: x = a + k m}$ $\ds$ $=$ $\ds \set {\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \ldots}$

The equivalence class $\eqclass a m$ is called the residue class of $a$ (modulo $m$).

## Examples

### Modulo $7$

The residue class of $2$ modulo $7$ on the integers is:

$\eqclass 2 7 = \set {\ldots, -19, -12, -5, 2, 9, 16, \ldots}$

## Also defined as

In their definition of a residue class, some sources are lax at defining $m$ as a strictly positive integer, and the restriction becomes clear only during further development of the theory.

Some sources with particular aims in mind are deliberately explicit about specifying that $m > 1$.

## Also known as

Residue classes are sometimes known as congruence classes (modulo $m$).

## Also see

• Results about residue classes can be found here.