# How many divisors?

**Number Theory**Level 3

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 \(1, 2, 3, 4, 6, 12\).