Congruence to an Integer less than Modulus

Theorem

Let $m \in \Z$.

Then each integer is congruent (modulo $m$) to precisely one of the integers $0, 1, \ldots, m - 1$.

Proof

Proof of Existence

Let $a \in \Z$.

Then from the Division Theorem: $\exists r \in \left\{{0, 1, \ldots, m-1}\right\}: a \equiv r \pmod m$.

Proof of Uniqueness

Suppose that:

$\exists r_1, r_2 \in \left\{{0, 1, \ldots, m-1}\right\}: a \equiv r_1 \pmod m \land a \equiv r_2 \pmod m$

Then:

$\exists r_1, r_2 \in \Z: a = q_1 m + r_1 = q_2 m + r_2$

This contradicts the uniqueness clause in the Division Theorem.

$\blacksquare$

Sources

• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 14.2 \ \text{(ii)}$