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 $a = b$.
$\blacksquare$
Sources
- 1971: George E. Andrews: Number Theory ... (previous) ... (next): $\text {4-1}$ Basic Properties of Congruences: Theorem $\text {4-3}$