\begin{definition}[Definition:Remainder]
Let $a, b \in \Z$ be integers such that $b \ne 0$.
From the Division Theorem, we have that:
:$\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$
The value $r$ is defined as the '''remainder of $a$ on division by $b$''', or the '''remainder of $\dfrac a b$'''.
\end{definition}