NP Complete Subset GCD Proof
Hint: Associate each edge $e$ with a prime $p_e$, and each vertex $v$ with a number $n_v \in S$. Choose $n_v$ so that the GCD of $\{n_v : v \in A\}$ encodes the set of edges covered by the vertices in $A$.

If you get stuck, try first showing that SubsetLCM is NP-complete. (LCM is Least Common Multiple.)