Problem: Suppose $p$ and $q$ are distinct prime numbers. How many subgroups of index $p$ are there in $\mathbb{Z} \times \mathbb{Z}$? And how many subgroups of index $pq$ are there in $\mathbb{Z} \times \mathbb{Z}$?
Attempt: I noted that $\mathbb{Z}\times \mathbb{Z}p < \mathbb{Z} \times \mathbb{Z}$ and $\mathbb{Z} \times \mathbb{Z} \mathrel{/} \mathbb{Z}\times \mathbb{Z}p=\mathbb{Z}_p$. Thus I would like to answer the first question with $2$ but I actually feel that it is wrong. Any help?