If x and y are positive integers for which 3x + 2y + xy = 115 + 14x + 4y - 2x + 5y + 3, then what is x + y?
+2668 Simplify to: \(-9x-7y+xy=118\)
Factor out the y: \(y(-7+x)-9x=118\)
Add 63 to both sides so we can apply Simon's favorite factoring trick: \(y(-7+x)-9x+63=118\)
Furthur factor: \((y-9)(-7+x)=181\)
181 is prime, so the only factors are 1 and 181
There are 2 cases for x and y for which the equation is satisfied: \((8,190)\) and \((10, 188)\)
Coincidentally, both sum to \(\color{brown}\boxed{198}\)
