jmc

First, we find $A$. The prime factorization of $500$ is $2^2\cdot 5^3$. Therefore, $$A=(1+2+2^2)(1+5+5^2+5^3)=(7)(156).$$To see why $(1+2+2^2)(1+5+5^2+5^3)$ equals the sum of the divisors of 500, note that if you distribute (without simplifying), you get 12 terms, with each divisor of $2^2\cdot 5^3$ appearing exactly once.

Now we prime factorize $7\cdot 156=7\cdot 2^2\cdot 3\cdot 13$. The sum of the prime divisors of $A$ is $2+3+7+13=\boxed{25}$.