\begin{definition}[Definition:Residue Class]
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:
{{begin-eqn}}
{{eqn | l = \eqclass a m
      | r = \set {x \in \Z: a \equiv x \pmod m}
      | c = 
}}
{{eqn | r = \set {x \in \Z: \exists k \in \Z: x = a + k m}
      | c = 
}}
{{eqn | r = \set {\ldots, a - 2 m, a - m, a, a + m, a + 2 m, \ldots}
      | c = 
}}
{{end-eqn}}
The equivalence class $\eqclass a m$ is called the '''residue class of $a$ (modulo $m$)'''.
\end{definition}