How many divisors?

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

This problem is posed by Michael T.

Details and assumptions

You may read up on Divisors of an Integer.

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

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