This challenge looks familiar - I think I answered something similar on our sister site, [Programming Puzzles and Code Golf](//codegolf.stackexchange.com).

There's a couple of pieces of good news:

1. You don't need to count 2's in the prime factorization.  There will always be more 2's than 5's, so you can never have an unpaired 5.
1. You don't need to iterate over every number up to `num` if you see that we're just computing `n/5 + n/25 + n/125 + ...` (integer division).  If we define `f(n) = n/5 + f(n/5)` and `f(0) = 0`, we have a much faster version.  That can be implemented recursively, or fairly easily converted to iterative form.