jmc

Let our numbers be $a$ and $b$ with $a>b.$ Then $ab+a+b=103$. With Simon's Favorite Factoring Trick in mind, we add $1$ to both sides and get $ab+a+b+1=104$, which factors as $(a+1)(b+1)=104$. We consider pairs $(a+1,b+1)$ of factors of $104$: $(104,1),(52,2),(26,4),(13,8)$. Since $a<20$, we can rule out the first 3 pairs, which gives $a=12$ and $b=7$, so $a+b=\boxed{19}$.