Problem
ALG-B1-M03-P019 Integer Solutions with a Bound
#19
★★★★☆ Level 4 of 5
Find all positive integers \(x,y\) such that \(xy=x+y+5\).
Move everything to the left and add \(1\) to get a product.
We have \(xy-x-y=5\). Add \(1\): \((x-1)(y-1)=6\).
Since \(x,y\) are positive integers, \(x-1\) and \(y-1\) are positive divisors of \(6\). The pairs \((1,6)\), \((2,3)\), \((3,2)\), \((6,1)\) give \((x,y)=(2,7),(3,4),(4,3),(7,2)\).
A classic product-completion trick.