On isomorphisms of the group of unit modulo n

Question

Let $U(n)$ denote the group of units of $\mathbb{Z}/n\mathbb{Z}$. I know that if $i$ and $j$ are relatively prime then $U(ij)$ is isomorphic to $U(i)\oplus U(j)$. I was wondering if the converse is true, namely if $U(ij)$ is isomorphic to $U(i)\oplus U(j)$ does this imply that $i$ and $j$ are relatively prime?

Answer 1

The order of $U(n)$ is $\varphi(n)$ and we have $\varphi(ij)=\varphi(i)\varphi(j)$ if and only if $i$ and $j$ are coprime.

to see this notice that $\varphi(n)=n\prod\limits_{p|n} \frac{p-1}{p}$ so if a prime is repeated we have $\phi(ij)> \phi(i)\phi(j)$