Problem
NT-B2-M07-P001 Completing a Product
#1
★★☆☆☆ Level 2 of 5
Find all positive integer pairs \((x,y)\) satisfying \(xy+2x+3y=54\).
Add \(6\) to both sides.
We have \(xy+2x+3y+6=60\), hence \((x+3)(y+2)=60\). Since \(x,y>0\), we need \(x+3\ge 4\), \(y+2\ge 3\). The suitable divisor pairs of \(60\) are \((4,15),(5,12),(6,10),(10,6),(12,5),(15,4),(20,3)\). Thus \((x,y)=(1,13),(2,10),(3,8),(7,4),(9,3),(12,2),(17,1)\). Each pair checks in the original equation.
A basic Simon's favorite factoring trick problem; suitable right after the theory.