**Corollary:** If a and b are integers, with b≠0, then there exists unique integers q and r such that a = q b + r; 0 ≤ r < |b|.

**Proof:** It is enough to consider the case in which b is negative. Then |b|>0 and the division algorithm theorem produces unique integers q′ and r for which a = q′ |b| +r, 0 ≤ r < |b|.

*We note that |b| = -b; we may take q = -q to arrive at a = q b + r,*

## with 0 ≤ r <|b|. (Proved)