If $\displaystyle A$ and $\displaystyle B$ are finite abelian groups of relatively prime orders, prove that $\displaystyle A \otimes_{\mathbb{Z}} B=0$. If $\displaystyle p$ is prime and $\displaystyle r>s$, prove $\displaystyle (\mathbb{Z}/p^r\mathbb{Z})\otimes_{\mathbb{Z}}(\mathbb{Z}/p^s\mathbb{Z})=\mathbb{Z}/p^s\mathbb{Z}$.

Any hints would be great (I haven't gotten far with this one!). Thanks in advance.