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Home / Elementary Number Theory / 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|.

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|.

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)

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