jmc

Let's find the cycle of the final three digits of $5^n$, starting with $n=3$ : $125,625,125,625,\dots$ . The cycle of the final three digits of $5^n$ is 2 numbers long: 125, 625. Thus, to find the final three digits of $5^n$ for any positive $n\ge 3$, we must find the remainder, $R$, when $n$ is divided by 2 ($R=1$ corresponds to 125, and $R=0$ corresponds to 625). Since $100÷2=50$ without remainder, the final three digits of $5^{100}$ are 625. Their sum is $6+2+5=\boxed{13}$.