jmc

We let our integer be $n$. the first sentence tells us that $$n\equiv 7(\mathrm{mod}15).$$ Since 3 and 5 are both factors of 15 we deduce $$\begin{array}{rl}n& \equiv 7\equiv 1(\mathrm{mod}3)\\ n& \equiv 7\equiv 2(\mathrm{mod}5).\end{array}$$ Therefore the remainders in question are 1 and 2, and their sum is $\boxed{3}$.