Problem
NT-B1-M05-P011 A Sum of Reciprocals
#11
★★☆☆☆ Level 2 of 5
Find all positive integers \(x,y\) such that \(\frac{1}{x}+\frac{1}{y}=\frac{1}{8}\).
Multiply by \(8xy\) and complete a product.
From \(8x+8y=xy\), we get \(xy-8x-8y=0\), hence \((x-8)(y-8)=64\). Therefore \(x=8+d\), \(y=8+\frac{64}{d}\), where \(d\) runs over the positive divisors of \(64\).
The student may also be asked to list all seven pairs explicitly.