$31 \cdot 9$ The number of positive integer solutions to the equation $2 x+3 y=763$ is
(A) 255.
(B) 254.
(C) 128.
(D) 127.
(E) 0.
(14th American High School Mathematics Examination, 1963)
[Solution] Let $x=\frac{763-3 y}{2}$. Since $x$ is a positive integer, $763-3 y$ must be even.
Therefore, $y$ must be a positive odd number, and $3 y \leqslant 763$.
There are 254 values of $y$ that satisfy the inequality, half of which are odd, i.e., 127. Hence, the answer is $(D)$.