1/x+1/y+1/xy=1/2007正整数（x,y）解个数为

Question

1/x+1/y+1/xy=1/2007 正整数（x,y）解个数为

Answer 1

快速的方法还没想到,先来个麻烦点的.
1/x + 1/y + 1/xy = 1/2007
(x + y + 1) / (xy) = 1/2007
2007x + 2007y + 2007 - xy = 0
(2007 - x)y = -2007x - 2007
y = 2007[(x+1)/(x-2007)] -------------------------> 得知 x > 2007
y = 2007[1 + (2008)/(x-2007)]
这样,若x,y 为正整数则 2008/(x-2007) 必须是正整数.
2008 = 2(2)(2)(251)
则 x-2007 = 1, 2, 4, 8, 251, 502, 1004, 2008
x = 2008, 2009, 2011, 2015, 2258, 2509, 3011, 4015
y = 4032063, 2017035, 1009521, 505764, 18063, 10035, 6021, 4014
x, y 值可以调换.