Division Theorem/Half Remainder Version

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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.