$31 \cdot 10$ The number of positive integer solutions to the equation $4 x+5 y=98$ is
(A) 4.
(B) 5.
(C) 6.
(D) infinitely many.
(China Junior High School Mathematics Correspondence Competition, 1987)
[Solution] Since $4 x+5 y=98$, then $y$ is even, and $y \leqslant 19$. Therefore, the possible values of $y$ are
$$
2,4,6,8,10,12,14,16,18 .
$$

Furthermore, when $y$ is a multiple of 4, the left side of the equation is a multiple of 4, but the right side 98 is not a multiple of 4. Therefore, $y$ must be of the form $4 k+2$, so
$$
y=2, \quad 6, \quad 10, \quad 14, \quad 18 \text {. }
$$

Checking each one, the corresponding $x$ values are all positive integers, so the equation $4 x+5 y=98$ has 5 sets of positive integer solutions.
Therefore, the answer is $(B)$.