Prove that if a and b are integers, then there are unique integers q and r such that $a = bq + r$, $-|b|/2 < r \le |b|/2$ Prove that if a and b are integers, then there are unique integers q and r such that $$a = bq + r,$$ with the restriction that$$-|b|/2 < r \le |b|/2$$

$r$ is the remainder after $b$ is divided by $qa$. If the remainder is larger than $a/2$ then increase $q$ by 1 and the remainder will become negative, since $r$ can't be more than half a multiple and less than half a multiple. Same concepts apply for negative numbers. 
