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(75p2)?
To find the minimum possible value of f(75p^2), we need to determine the smallest number of distinct positive factors that 75p^2 can have.
First, we need to determine the prime factorization of 75p^2. We know that 75 = 3 * 5 * 5, and since p is a prime number, p^2 is the prime factorization of p raised to the power of 2.
Therefore, the prime factorization of 75p^2 is 3 * 5 * 5 * p * p.
To find the number of distinct positive factors, we can consider the powers of the prime factors. Each power can vary from 0 to the power raised. For example, the factorization of 75p^2 can be written as (3^a) * (5^b) * (p^c), where a, b, and c are non-negative integers.
Now, let's analyze the possibilities for each power:
1. For the power of 3: The possible values are 0 or 1 because 3^2 > 75p^2. So we have two possibilities: (3^0) or (3^1).
2. For the power of 5: The possible values are 0, 1, or 2 because 5^3 = 125 > 75p^2. So we have three possibilities: (5^0), (5^1), or (5^2).
3. For the power of p: The possible values are 0, 1, or 2 because p^3 > 75p^2. So we have three possibilities: (p^0), (p^1), or (p^2).
To find the minimum possible value of f(75p^2), we need to choose the smallest possible power for each prime factor. In this case, (3^0) * (5^0) * (p^0) represents the number 1, which doesn't have any other distinct factors besides itself.
Therefore, the minimum possible value of f(75p^2) is 1.