Let $R = (\mathbb{Z}, +, \cdot)$ be the D588: Ring of integers.
Let $a, b \in \mathbb{Z}$ each be a D5094: Positive integer such that

(i)	\begin{equation} a \neq 0 \end{equation}

Then \begin{equation} \# \{ (q, r) \in \mathbb{Z} \times \mathbb{Z} : b = a q + r \text{ and } 0 \leq r < |a| \} = 1 \end{equation}

Let $R = (\mathbb{Z}, +, \cdot)$ be the D588: Ring of integers.
Let $a, b \in \mathbb{Z}$ each be a D5094: Positive integer such that

(i)	\begin{equation} a \neq 0 \end{equation}

Then \begin{equation} \exists \, ! \, (q, r) \in \mathbb{Z} \times \mathbb{Z} : b = a q + r \text{ and } 0 \leq r < |a| \end{equation}