smc

By Fermat's Little Theorem, we know that $2^{100}\equiv 2^{1000(\mathrm{mod}12)}(\mathrm{mod}13)$. However, we find that $1000\equiv 4(\mathrm{mod}12)$, so $2^{1000}\equiv 2^4=16\equiv 3(\mathrm{mod}13)$, so the answer is $\boxed{3}$.