Let S be a nonempty subset of $\mathbb{Z}$. Suppose S satisfies the following constraints

If $x,y \in S$ then $x+y \in S $

If $x \in S$ and $y \in \mathbb{Z}$ then $xy \in S$

Show that S is the set of all integer multiples of m for some $m \in \mathbb{N}\cup{\{0\}} $

So far I can think of a few things that might be useful, this might be a proof by cases where we split it up into the case where $ S\cap\mathbb{N} =\emptyset$ and $ S\cap\mathbb{N} \neq\emptyset$ which is when $m=0,m\neq 0$ respectively. But how do I use the constraints to come to that conclusion? I don't need a full solution but a hint would be nice. Thanks!