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:

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

Also defined as
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

 * Congruent Integers in Same Residue Class


 * Definition:Set of Residue Classes