If $x$ and $y$ are positive numbers less than $20$ for which $x+y+xy=76$, what is $x+y$?

Question

What is a simple way to solve this problem? I can do it by trying $x$ and $y$, starting from $1$. That does not look like the best way.

If $x$ and $y$ are positive numbers less than $20$ for which $x+y+xy=76$, what is $x+y$?

Answer 1

\begin{align}xy+x+y&=76\\ x(y+1)+y&=76\\ x(y+1)+y+1&=77\\ (x+1)(y+1)&=77 \end{align}

As $1,7,11,77$ are the only positive divisors of $77$, $(x,y)$ should be $(6,10)$ or $(10,6)$. Thus, $x+y=16$

Answer 2

hint:$x+y+xy=76\implies x+y+xy+1=77\implies (x+1)(y+1)=77$