31.48 The number of positive integer solutions to the system of equations $\left\{\begin{array}{l}x y+y z=63, \\ x z+y z=23\end{array}\right.$ is 
(A)1.
(B) 2.
(C) 3.
(D) 4.
(China Junior High School Mathematics League, 1995)
[Solution] The 2nd equation is $(x+y) z=23$. Since $x+y \geqslant 2$, and 23 is a prime number, we have $z=1, x+y=23$.
Therefore, $y=23-x$, substituting this into the 1st equation, we get
$$
(23-x)(x+1)=63 \text {, }
$$

which simplifies to
$$
x^{2}-22 x+40=0 \text {. }
$$

Solving this, we get $x_{1}=2, x_{2}=20$, so the two sets of solutions to the system of equations are:
$$
\begin{array}{l}
x_{1}=2, \quad y_{1}=21, \quad z_{1}=1 \text {; } \\
x_{2}=20, \quad y_{2}=3, \quad z_{2}=1 ; \\
\end{array}
$$

Therefore, the answer is $(B)$.