I've seen the proof that for gcd$(m,n)=1$ the product $\mathbb{Z}/{n \mathbb{Z}} \times \mathbb{Z}/{m \mathbb{Z}}$ is cyclic since it is isomorphic to $\mathbb{Z}/{nm \mathbb{Z}}$ by the Chinese Remainder Theorem.
But now I am asked to show that the external direct sum $\mathbb{Z}/{n \mathbb{Z}} \oplus \mathbb{Z}/{m \mathbb{Z}}$ is cyclic iff gcd$(m,n)=1$ and I'm not sure why this is true. It seems to be related enough but I could do with some help.