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)?
To find the minimum possible value of f(75p^2), we need to determine the prime factorization of 75p^2 first.
The prime factorization of 75 is: 3 * 5^2
Since p is a prime number, we can say that p = p^1
So, the prime factorization of 75p^2 becomes: 3 * 5^2 * p^2
To find the number of distinct positive factors, we need to consider the exponents of each prime factor. The number of factors will be the product of the incremented exponents:
(1+1) * (2+1) * (2+1) = 2 * 3 * 3 = 18
Hence, the minimum possible value of f(75p^2) is 18.