Common Factor Cancelling in Congruence

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

Let:
 * $$a x \equiv b y \left({\bmod\, m}\right)$$ and $$a \equiv b \left({\bmod\, m}\right)$$

where $$a \equiv b \left({\bmod\, m}\right)$$ denotes that $$a$$ is congruent modulo $m$ to $$b$$.

Then:
 * $$x \equiv y \left({\bmod\, \frac m d}\right)$$

where $$d = \gcd \left\{{a, m}\right\}$$.

Corollary
If $$a$$ is coprime to $$m$$, then:
 * $$x \equiv y \left({\bmod\, m}\right)$$

Proof
We have that $$d = \gcd \left\{{a, m}\right\}$$.

From Law of Inverses (Modulo Arithmetic), we have:
 * $$\exists a' \in \Z: a a' \equiv d \left({\bmod\, m}\right)$$

Hence:

$$ $$

Then:

$$ $$ $$ $$

Hence the result.

Proof of Corollary
If $$a \perp m$$ then $$\gcd \left\{{a, m}\right\} = 1$$ by definition.

The result is immediate.