Integer is Congruent Modulo Divisor to Remainder

Theorem

Let $a \in \Z$.

Let $a$ have a remainder $r$ on division by $m$.

Then:

$a \equiv r \pmod m$

where the notation denotes that $a$ and $r$ are congruent modulo $m$.

Corollary

$a \equiv b \pmod m$ if and only if $a$ and $b$ have the same remainder when divided by $m$.

Proof

Let $a$ have a remainder $r$ on division by $m$.

Then:

$\exists q \in \Z: a = q m + r$

Hence by definition of congruence modulo $m$:

$a \equiv r \pmod m$

$\blacksquare$

Sources

• 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 14.2 \ \text{(i)}$: Congruence modulo $m$ ($m \in \N$)