Congruence (Number Theory) is Congruence Relation

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Theorem

Congruence modulo $m$ is a congruence relation on $\left({\Z, +}\right)$.


Proof

Suppose $a \equiv b \bmod m$ and $c \equiv d \bmod m$.

Then by the definition of congruence there exists $k, k' \in \Z$ such that:

$\left({a - b}\right) = km$
$\left({c - d}\right) = k'm$

Hence:

$\left({a - b}\right) + \left({c - d}\right) = km + k'm$

Using the properties of the integers:

$\left({a + c}\right) - \left({b + d}\right) = m \left({k + k'}\right)$


Hence $\left({a + c}\right) \equiv \left({b + d}\right) \bmod m$ and congruence modulo $m$ is a congruence relation.

$\blacksquare$


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