What is the smallest prime number dividing the sum $3^{11}+5^{13}$?
$\mathrm{(A)\ } 2 \qquad \mathrm{(B) \ }3 \qquad \mathrm{(C) \  } 5 \qquad \mathrm{(D) \  } 3^{11}+5^{13} \qquad \mathrm{(E) \  }\text{none of these}$
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 $2, \boxed{\text{A}}$.