2012 AMC 12A Problem 17

Let S be a subset of \{1,2,3,\ldots,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\qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 18

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.