If $n > 2$, prove that the order of the multiplicative group of units modulo n, $U_n$, is even.

Hint: Use a pairing argument. If $x$ is a unit, then so is $-x$. And if $n\gt 2$ they are distinct.

Yet another hint: Show that $\{+1,-1\}$ is a subgroup of $U_n$ of order $2$. Now apply Lagrange.