AMC 12 2012a Problem 17

Let $S$ be a subset of $\{1,2,3,\dots,30\}$ with the property that no pair of distinct elements in $S$ has a sum divisible by $5$. What is the largest possible size of $S$?

\textbf{(A)}\ 10 \textbf{(B)}\ 13 \textbf{(C)}\ 15 \textbf{(D)}\ 16 \textbf{(E)}\ 18