This is equivalent to: Characterize all submonoids of the monoid $\mathbb{N}_{0}$. Proof: Let $a_{1}, a_{2}\in S$. Then all big enough multiples $m$ of $c_{1}= \gcd(a_{1}, a_{2})$ are of type $m=xa_{1}+ya_{2}$ with $x,y \geq 0$, thus are in $S$. Now if there is an $a_{3}$ that is not a multiple of $\gcd(a_{1}, a_{2})$, we procede to same way to get that all big enough multiples of $\gcd(a_{1},a_{2},a_{3})$ are in $S$. We do this again and again. It stops at some point since the number of different residues $\mod \gcd(a_{1},a_{2},a_{3},..., a_{i})$ strictly decreases.