Common Factor Cancelling in Congruence/Corollary 2

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Corollary to Common Factor Cancelling in Congruence

Let $a, b, x, y, m \in \Z$.

Let:

$a x \equiv a y \pmod m$

where $a \equiv b \pmod m$ denotes that $a$ is congruent modulo $m$ to $b$.


If $a$ is coprime to $m$, then:

$x \equiv y \pmod m$


Proof

From Common Factor Cancelling in Congruence: Corollary 1:

$a x \equiv b y \pmod m$ and $a \equiv b \pmod m$

and:

$a$ is coprime to $m$

then:

$x \equiv y \pmod m$


The result follows immediately from setting $a = b$.

$\blacksquare$


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