$32 \cdot 47$ Let $f(x)=x^{2}+3 x+2$, and let $S=\{0,1,2, \cdots, 25\}$. If $s \in S$, and $f(s)$ is a multiple of 6, then the number of such $s$ is 
(A) 25.
(B) 22.
(C) 21.
(D) 18.
(E) 17.
(15th American High School Mathematics Examination, 1964)
[Solution] Since $x^{2}+3 x+2=(x+2)(x+1)$, then in $S$ the elements that make $s+2$ a multiple of 6 are
$$
s=4+6 k \quad(k=0,1,2,3).
$$

Also, in $S$ the elements that make $s+1$ a multiple of 6 are
$$
s=5+6 k \quad(k=0,1,2,3).
$$

Furthermore, in $S$ the elements that make $s+2$ a multiple of 3 and $s+1$ even, or $s+2$ even and $s+1$ a multiple of 3 are
the former $s=1+6 k \quad(k=0,1,2,3,4)$.
the latter $s=2+6 k \quad(k=0,1,2,3)$.

In summary, there are 17 elements in $S$ that make $f(s)$ a multiple of 6:
$$
\begin{array}{ll}
4,10,16,22 ; & 5,11,17,23 ; \\
2,8,14,20 ; & 1,7,13,19,25 .
\end{array}
$$

Therefore, the answer is $(E)$.