$31 \cdot 8$ The number of positive integer solutions to the equation $3 x+5 y=501$ is
(A) 33.
(B) 34.
(C) 35.
(D) 100.
(E) None of the above.
(18th American High School Mathematics Examination, 1967)
[Solution] From the given, we have $y=\frac{3(167-x)}{5}$. Since $y$ is a positive integer, then $167-x$ is a multiple of 5. Thus, $\quad x=5 k+2, k=0,1,2, \cdots, 32$. Therefore, the answer is $(A)$.