How many divisors?

For a positive integer xx, let f(x)f(x) be the function which returns the number of distinct positive factors of xx. If pp is a prime number, what is the minimum possible value of f(75p2)?f(75p^2)?

This problem is posed by Michael T.

Details and assumptions

You may read up on Divisors of an Integer.

The distinct positive factors of xx need not be prime. For example, f(12)=6 f(12) = 6, since 12 has distinct positive factors of 1,2,3,4,6,121, 2, 3, 4, 6, 12.

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