Definition:Multiplicative Order of Integer

Context
Number Theory

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 $$a \perp n$$.