$U(pq)$ is to be the units of $Z_{pq}$. I would assume that distinct prime factors would enable $U(pq)$ to always by cyclic. My professor alluded to the fact that this may not always be true. However, I am curious if this is always the case, since $U(p)$ is cyclic, given $p$ is prime.
What are your thoughts?
If $U(15)=U(3×5)$ were cyclic, there would be a primitive root modulo $15$, but there is none. Thus $U(15)$ is not cyclic.
$U(n)$ is cyclic iff $n=1,2,4,p^k,2p^k$ where $p$ is an odd prime and $k\ge1$.