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


\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, $x+y=16$
