Definition:Multiplicative Order of Integer

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

Then the order of $a$ modulo $n$ is the least positive integer $c$ such that:
 * $a^c \equiv 1 \left({\bmod\, n}\right)$

Such a number $c$ exists iff $a$ and $n$ are coprime.

Conditions on Existence of the Order of an Integer
From Integer has Order Modulo n iff Coprime to n it is seen that it is necessary for $a \perp n$.

From Euler's Theorem it is sufficient for $a \perp n$.