Division Theorem

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 $0 \le r < \left|{b}\right|$:


 * $\forall a, b \in \Z, b \ne 0: \exists! q, r \in \Z: a = q b + r, 0 \le r < \left|{b}\right|$

In the above equation:
 * $q$ is called the quotient
 * $r$ is called the principal remainder, or, more usually, just the remainder.

Also known as
Otherwise known as the Quotient Theorem, or (more specifically) the Quotient-Remainder Theorem (as there are several other "quotient theorems" around).

Some sources call this the division algorithm but it is preferable not to offer up a possible source of confusion between this and the Euclidean Algorithm to which it is closely related.

It is also known by some as Euclid's Division Lemma.