Problem
NT-B1-M05-P013 A Mixed Product
#13
★★★☆☆ Level 3 of 5
Find all positive integer solutions of \(xy+2x+3y=54\).
Add \(6\) to obtain a product.
We have \(xy+2x+3y+6=60\), so \((x+3)(y+2)=60\). Since \(x\ge1\), \(y\ge1\), we need \(x+3\ge4\), \(y+2\ge3\). Checking the divisors of \(60\), we get \((1,13)\), \((2,10)\), \((3,8)\), \((7,4)\), \((9,3)\), \((12,2)\), \((17,1)\).
It is convenient to set \(a=x+3\), \(b=y+2\).