20 Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation
$$
x \sqrt{y}+y \sqrt{x}+\sqrt{2009 x y}-\sqrt{2009 x}-\sqrt{2009 y}-2009=0 .
$$
20 Answer: (6)
$$
\begin{array}{l}
\text { Note that } x \sqrt{y}+y \sqrt{x}+\sqrt{2009 x y}-\sqrt{2009 x}-\sqrt{2009 y}-2009 \\
=(\sqrt{x}+\sqrt{y}-\sqrt{2009})(\sqrt{x y}-\sqrt{2009}), \text { so } x y=2009 . \\
\therefore(x, y)=(1,2009),(7,287),(41,49),(49,41),(287,7),(2009,1) .
\end{array}
$$