20. Determine the number of ordered pairs $(x, y)$ of positive integers satisfying the following equation:求滿足下式的正整數序偶對 $(x, y)$ 的數目:
$$
x \sqrt{y}+y \sqrt{x}-\sqrt{2007 x}-\sqrt{2007 y}+\sqrt{2007 x y}=2007
$$
20. 6
20. Rewrite the equation as $x \sqrt{y}+y \sqrt{x}+\sqrt{2007 x y}=\sqrt{2007 x}+\sqrt{2007 y}+2007$, which becomes $\sqrt{x y}(\sqrt{x}+\sqrt{y}+\sqrt{2007})=\sqrt{2007}(\sqrt{x}+\sqrt{y}+\sqrt{2007})$ upon factorisation. Since the common factor $\sqrt{x}+\sqrt{y}+\sqrt{2007}$ is positive, we must have $x y=2007=3^{2} \times 223$. As 2007 has $(2+1)(1+1)=6$ positive factors, the answer is 6 .