Integer is Congruent to Integer less than Modulus

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Let $m \in \Z$.

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


Proof of Existence

Let $a \in \Z$.

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

Proof of Uniqueness

Suppose that:

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


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

This contradicts the uniqueness clause in the Division Theorem.