31. How many ordered pairs of positive integers $(x, y)$ satisfy the equation
$$
x \sqrt{y}+y \sqrt{x}+\sqrt{2006 x y}-\sqrt{2006 x}-\sqrt{2006 y}-2006=0 ?
$$
31. Ans: 8
$$
\begin{aligned}
& x \sqrt{y}+y \sqrt{x}+\sqrt{2006 x y}-\sqrt{2006 x}-\sqrt{2006 y}-2006=0 \\
\Longleftrightarrow & (\sqrt{x}+\sqrt{y}+\sqrt{2006})(\sqrt{x y}-\sqrt{2006})=0 \\
\Longleftrightarrow & \sqrt{x y}-\sqrt{2006}=0 \quad(\text { since } \sqrt{x}+\sqrt{y}+\sqrt{2006}>0) \\
\Longleftrightarrow & x y=2006=17 \times 59 \times 2 .
\end{aligned}
$$

Thus the solutions are
$$
(1,2006),(2006,1),(2,1003),(1003,2),(34,59),(59,34),(17,118),(118,17) .
$$