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 minimum number of distinct positive factors that 75p^2 can have.
First, let's factorize 75p^2.
The prime factorization of 75 is 3^1 * 5^2.
And the prime factorization of p^2 is p * p.
To find the number of distinct positive factors, we need to consider all possible combinations of the prime factors. Each factor can have an exponent of 0 to the maximum power of each prime factor.
For 75p^2, we have 3 * 5^2 * p * p.
The exponents of 3, 5, and p can be:
3: 0, 1
5: 0, 1, 2
p: 0, 1, 2
To get the minimum number of distinct positive factors, we should choose the smallest possible exponent for each prime factor.
Therefore, the minimum possible value of f(75p^2) occurs when all the exponents are 0:
f(75p^2) = (0 + 1) * (0 + 1) * (0 + 1) = 1 * 1 * 1 = 1.
So, the minimum possible value of f(75p^2) is 1.