31 - 44 The system of equations $\left\{\begin{array}{l}x+2 y=n \\ 3 x-y=m\end{array}\right.$ (where $n, m$ are both odd numbers) has integer solutions, then the solution $\left\{\begin{array}{l}x=p \\ y=q\end{array}\right.$ has the parity situation:
(A) $p, q$ are both even.
(B) $p, q$ are both odd.
(C) $p$ is even, $q$ is odd.
(D) $p$ is odd, $q$ is even.
(2nd China Tianjin "Chinese Youth Cup" Mathematics Invitational Competition, 1988)
［Solution］From the given system of equations, we can solve to get
$$
\left\{\begin{array}{l}
x=\frac{2 m+n}{7}, \\
y=\frac{3 n-m}{7} .
\end{array}\right.
$$
$\because m, n$ are both odd numbers,
$\therefore 2 m+n$ is an odd number. Since 7 is also an odd number, and $x$ is an integer, then $x=p$ is an odd number. Similarly, we can deduce that $y=q$ is an even number.
Therefore, the answer is $(D)$.