Congruence to Remainder

Theorem

If $a \in \Z$ has a remainder $r$ on division by $m$, then $a \equiv r \pmod m$.

Corollary

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

Proof

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

Then $\exists q \in \Z: a = qm + r$.

Hence $a \equiv r \pmod m$.

$\blacksquare$

Proof of Corollary

Follows directly from the above and Congruence Modulo m Equivalence.

$\blacksquare$

Sources

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