I'm struggling with this. I know it is going to use Lagrange's Theorem. This is what I have so far.
Suppose $|U_n| = k.$ This implies $a^k = 1$ for all $a$ in $U_n$ and $|a|$ divides $k$.
Now, what can be said about the order of an element $a$? I haven't been able to conclude that it's even with what I have, therefore I continued so: $|\langle a\rangle|$ divides $k$ where $\langle a\rangle$ is the cyclic subgroup generated by $a.$
However, I haven't been able to conclude that $|\langle a\rangle|$ is even.
So I tried using this: an element $a$ is in $U_n$ iff gcd$(a, n) = 1$.