The order of an element

Question

The order of a unit $a \pmod m $ is the least $n \geq 1$ such that $a^n \equiv 1 \pmod m$.
my question is : Is true that number and its inverse have the same order?

Answer 1

Suppose not. Then what can you say about $a^n (a^{-1})^n \pmod m$?

Answer 2

Notice that we have $\ a^n \equiv 1 \iff (a^{-1})^n \equiv 1\ $ since $\ a^n (a^{-1})^n \equiv 1$

Thus $\ \{ n\ge 1 :\ a^n \equiv 1\} = \{ n\ge 1\ :\ (a^{-1})^n\equiv 1\}\ $ so they have equal least element = order.