Factorize the given number:
[tex]5^5 - 5^3 = 5^3 \left(5^2 - 1\right) = 5^3 \times 24 = 2^3 \times 3 \times 5^3[/tex]
The first equality follows from
[tex]5^5 = 5^{3 + 2} = 5^3 \times 5^2[/tex]
so both 5⁵ and 5³ share a common factor of 5³ that we can pull out as we did.
Then the distinct prime factors are 2, 3, and 5; their sum is 2 + 3 + 5 = 10.