13. Given that $m$ and $n$ are positive integers, the equation $x^{2}-m n x+(m+n)=0$ has positive integer solutions. Find the values of $m$ and $n$.
13. Let the equation $x^{2}-m n x+(m+n)=0$ have two roots $\alpha, \beta$. Then $\left\{\begin{array}{l}\alpha+\beta=m n, \\ \alpha \beta=m+n .\end{array}\right.$
Given that $m, n, \alpha, \beta$ are all positive integers, without loss of generality, assume
$$
\alpha \geqslant \beta \geqslant 1, m \geqslant n \geqslant 1 \text {. }
$$

Thus, $\alpha+\beta-\alpha \beta=m n-(m+n)$, which means $(\alpha-1)(\beta-1)+(m-1)(n-1)=2$. Therefore, $\left\{\begin{array}{l}(\alpha-1)(\beta-1)=2,1,0, \\ (m-1)(n-1)=0,1,2 .\end{array}\right.$
Solving this, we get $\left\{\begin{array}{l}\alpha=3,2,5, \\ \beta=2,2,1, \\ m=5,2,3 \\ n=1,2,2 .\end{array}\right.$
Therefore, $(m, n)=(5,1),(2,2),(3,2)$.