Problem
NT-B2-M07-P002 Two Unit Fractions
#2
★★☆☆☆ Level 2 of 5
Find all ordered pairs of positive integers \((x,y)\) such that \(\frac{1}{x}+\frac{1}{y}=\frac{1}{10}\).
After multiplying by \(10xy\), complete the product \((x-10)(y-10)\).
From \(10x+10y=xy\), we get \(xy-10x-10y=0\), hence \((x-10)(y-10)=100\). Thus \(x=10+d\), \(y=10+\frac{100}{d}\), where \(d\) runs over the positive divisors of \(100\). The answer is \((11,110),(12,60),(14,35),(15,30),(20,20),(30,15),(35,14),(60,12),(110,11)\).
Note that the pairs are ordered; for unordered pairs the answer would be shorter.