smc

Since we want to find the smallest prime dividing the sum, we start with the smallest prime and move up, so first we try $2$. Notice that $3^{11}$ and $5^{13}$ are both odd, so their sum must be even. This means that $2$ must divide $3^{11}+5^{13}$, and so since $2$ is the smallest prime, our answer must be $2,\boxed{2}$.