34. Consider the simultaneous equations
$$
\left\{\begin{array}{l}
x y+x z=255 \\
x z-y z=224 .
\end{array}\right.
$$

Find the number of ordered triples of positive integers $(x, y, z)$ that satisfy the above system of equations.
34. Ans: 2
Subtracting the second equation from the first one, we have
$$
\begin{aligned}
x y+y z=255-224=31 \\
\Longrightarrow y(x+z)=31 .
\end{aligned}
$$

Since 31 is a prime number, we have $y=1$ and $x+z=31$. Together with the first given equation, we have
$$
\begin{aligned}
& x \cdot 1+x(31-x)=255 \\
\Longrightarrow & (x-15)(x-17)=0 \\
\longrightarrow & x=15 \text { or } 17 .
\end{aligned}
$$

When $x=15, y=1$, we have $z=16$.
When $x=17, y=1$, we have $z=14$.
Therefore, there are two such solutions, namely $(15,1,16)$ and $(17,1,14)$.