smc

The sum of the first $n$ integers is given by $\frac{n(n+1)}{2}$, so $\frac{37(37+1)}{2}=703$. Therefore, $703-x-y=xy$ Rearranging, $xy+x+y=703$. We can factor this equation by SFFT to get $(x+1)(y+1)=704$ Looking at the possible divisors of $704=2^6\cdot 11$, $22$ and $32$ are within the constraints of $0<x\le y\le 37$ so we try those: $(x+1)(y+1)=22\cdot 32$ $x+1=22,y+1=32$ $x=21,y=31$ Therefore, the difference $y-x=31-21=\boxed{10}$.