Law of Inverses (Modulo Arithmetic)/Corollary 1

Corollary to Law of Inverses (Modulo Arithmetic)
Let $m, n \in \Z$ such that:
 * $m \perp n$

that is, such that $m$ and $n$ are coprime.

Then:
 * $\exists n' \in \Z: n n' \equiv 1 \pmod m$

Proof
By definition of coprime:
 * $m \perp n \iff \gcd \left\{{m, n}\right\} = 1$

The result follows directly from Law of Inverses (Modulo Arithmetic).