For a positive integer n, if $5^n$ is a factor of $25!$ , but $5n+1$ is not a factor of $25!$ , what is the value of n?

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What this question tests

Your command of divisibility, primes, factors, parity, and how the GMAT tests properties of integers.

Forgetting that 1 is not prime, that 0 is divisible by every integer, or that negative numbers flip your assumptions.

For a positive integer n, if 5n5^n5n is a factor of 25!25!25!, but 5n+15n+15n+1 is not a factor of 25!25!25!, what is the value of n?