Problem
NT-B1-M05-P015 Reciprocals with \(10\)
#15
★★★☆☆ Level 3 of 5
Find all positive integer solutions of \(\frac{1}{x}+\frac{1}{y}=\frac{1}{10}\).
After multiplying by \(10xy\), obtain \((x-10)(y-10)=100\).
Multiplying gives \(10x+10y=xy\). Then \(xy-10x-10y=0\), and \((x-10)(y-10)=100\). Hence \(x=10+d\), \(y=10+\frac{100}{d}\), where \(d\) is a positive divisor of \(100\). This describes all solutions.
One can discuss why negative divisors do not give positive \(x,y\).