jmc

Suppose that 300 has $d$ divisors. The divisors of 300 can be split into $d/2$ pairs in such a way that the product of each pair is 300: $\{1,300\},\{2,150\},$ etc. Thus $A=300^{d/2}$, which implies that $A$ has the same prime divisors as 300. Since the prime factorization of $300$ is $2^2\cdot 3\cdot 5^2$, the sum of the prime divisors of $A$ is $2+3+5=\boxed{10}$.