Law of Inverses (Modulo Arithmetic)

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

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

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

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

So:

So $b$ (in the above) fits the requirement for $n'$ in the assertion to be proved.