1964 AHSME Problem 16

Let f(x)=x^2+3x+2 and let S be the set of integers \{0, 1, 2, \ldots , 25 \}. The number of members s of S such that f(s) has remainder zero when divided by 6 is:

\textbf{(A)}\ 25\qquad \textbf{(B)}\ 22\qquad \textbf{(C)}\ 21\qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 17

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