$U(pq)$ 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?
Prove that $U_{pq}$ is not cyclic if $p$ and $q$ are distinct odd primes.   I am a self taught person. I just learned this and tried this on my own and came up with this.  $x \equiv 1 \pmod{p}$ and $x \equiv -1 \pmod{q}$ has a solution $[a]_{pq}$, since $p$ and $q$ are relatively prime. Because $q$ is an odd prime, $[-1]_{pq}$ is not a solution, so $[a]_{pq}\neq [-1]_{pq}$. But $a^2 \equiv 1 \pmod{p}$ and $a^2 \equiv 1 \pmod{q}$, so $a^2 \equiv 1\pmod{pq}$ since $p$ and $q$ are relatively prime, and thus $[a]_{pq}$ has order 2.  Can someone please tell me if this proof this correct? Please help with proof as I learned it just now as a self taught person.
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