Problem
NT-B2-M07-P007 A Fraction Equation With an Extra Term
#7
★★★☆☆ Level 3 of 5
Find all positive integer pairs \((x,y)\) if \(\frac{1}{x}+\frac{1}{y}+\frac{1}{xy}=\frac{1}{8}\).
Multiply by \(8xy\).
We get \(8x+8y+8=xy\), so \(xy-8x-8y=8\). Therefore \((x-8)(y-8)=72\). For every positive divisor \(d\mid 72\), \(x=8+d\), \(y=8+\frac{72}{d}\). The solutions are \((9,80),(10,44),(11,32),(12,26),(14,20),(16,17),(17,16),(20,14),(26,12),(32,11),(44,10),(80,9)\).
Extends the two-unit-fraction example: the extra term is harmless if multiplied carefully.