Division Theorem/Half Remainder Version

Theorem
For every pair of integers $a, b$ where $b \ne 0$, there exist unique integers $q, r$ such that $a = q b + r$ and $-\dfrac {\size b} 2 \le r < \dfrac {\size b} 2$:


 * $\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, -\dfrac {\size b} 2 \le r < \dfrac {\size b} 2$

In the above equation:
 * $a$ is the dividend
 * $b$ is the divisor
 * $q$ is the quotient
 * $r$ is the remainder.