Question 14
Let $S=\{1,2,3,4,\ldots,16\}$. Each of the subsets $\{6\}$, $\{1,2,3\}$, $\{5,7,9,10,11,12\}$, $\{1,2,3,4,5,6,7,8,9\}$ has sum divisible by $3$. Find the total number of nonempty subsets $A\subseteq S$ whose element sum is divisible by $3$.