I came across this statement in an abstract algebra textbook, I am looking for a proof.
EDIT: I guess I was not clear, what I meant was
If $k\ \epsilon\ U(n)$ and $m\ \epsilon\ \mathbb{Z}^+$ and
$k^m \equiv 1\ mod\ n$ ,
why is the smallest value of $m$ that satisfies the congruence for all $k$ equal to $\varphi(n) $ ?