# 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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