Integer has Multiplicative Order Modulo n iff Coprime to n

Theorem
Let $$a$$ and $$n$$ be integers.

Let $$c \in \Z_+$$ be the order of $a$ modulo $n$.

Then $$a \perp n$$, that is, $$a$$ and $$n$$ are coprime.

Proof
By definition, if $$c \in \Z_+$$ is the order of $a$ modulo $n$, then:
 * $$a^c \equiv 1 \left({\bmod\, n}\right)$$

Hence by definition, $$a^c = k n + 1$$.

Thus $$a r + n s = 1$$ where $$r = a^{c-1}$$ and $$s = -k$$.

The result follows from Integer Combination of Coprime Integers.