mutually exclusive sets closed under addition

Question

I was wondering if it is possible to find, or prove the existence of two sets S and S', both subsets of the natural numbers, that are mutually exclusive and respectively closed under addition. An example would be the trivial solution {0}, and all other numbers that are not zero. These sets are mutually exclusive and closed under addition.
Can anyone prove the existence/provide an example of/ disprove the existence of two nontrivial, mutually exclusive subsets of the natural numbers, or lead me to resources that help me further my knowledge?
Thanks!

Answer 1

Assuming that both subset don't have $0$. Because the two subsets $S_1,S_2$ are closed under addition, we have that they contain all products of their minimal elements. Let $a$ be the smallest element in $S_1$ and $b$ be the smallest element in $S_2$ then $aN=\{a,2a,3a...\} \subset S_1$ and $bN=\{b,2b,3b,...\} \subset S_2$. So then $S_1,S_2$ must both have $ab$.