Law of Inverses (Modulo Arithmetic)

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

Then:
 * $$\exists n' \in \Z: n n' \equiv d \left({\bmod\, m}\right)$$

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

Corollary
Let $$m, n \in \Z$$ such that $$m \perp n$$, i.e. such that $$m$$ and $$n$$ are coprime.

Then:
 * $$\exists n' \in \Z: n n' \equiv 1 \left({\bmod\, m}\right)$$

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.

Proof of Corollary
We have that $$m \perp n$$.

That is, $$\gcd \left\{{m, n}\right\} = 1$$.

The result follows directly.

Note
In the equivalence $$n n' \equiv 1 \left({\bmod\, m}\right)$$ note that from Euler's Theorem:
 * $$n' \equiv n^{\phi \left({n}\right) - 1} \left({\bmod\, m}\right)$$

where $$\phi \left({n}\right)$$ is the Euler $\phi$ function.