Integer Coprime to Modulus iff Linear Congruence to 1 exists

Theorem
Let $a, m \in \Z$.

The linear congruence:
 * $a x \equiv 1 \pmod m$

has a solution $x$ iff $a$ and $m$ are coprime.

Proof
From Integer Combination of Coprime Integers:
 * $a \perp m \iff \exists x, y \in \Z: a x + m y = 1$

That is, such an $x$ exists iff $a$ and $m$ are coprime.