For 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)?
I assume you mean
what is the minimum possible value of f(75p²).
Is it known if p is greater than 5?
If it is, would it be the number of distinct factors of 75 multiplied by 3 (6*3)=18, namely 1,3,5,15,25,75, then multiply each one by p, and then multiply each one by p².
no its not given that it is greater than 5 but it must be an prime number
Shame on you Keshav!!! Cheating on Brilliant!!! This site is meant to be a platform to practice your own skills, not to copy paste the questions and get free answers and then get incentives without effort. So either play fair and be honest or leave this site. People like you are shame to the Brilliant community. And to the others, please give the answer to this problem after Monday 10/6/2013, so that this cheat doesn't get the opportunity to cheat.
Do you know, Keshav, what you just did? You ruined the week long effort of Brillint Challenge Master Calvin by posting the problems here. If you can't solve the problems yourself why don't you just leave the site?
If you continue doing this, you will never be able to improve. The objective of the site is not to give prizes and get quick points but to sharpen one's problem solving skills.
To find the minimum possible value of f(75p^2), we need to first express the number 75p^2 in terms of its prime factorization.
Prime factorization of 75:
75 = 3 * 5 * 5
Prime factorization of p^2:
p^2 = p * p
Multiplying these two prime factorizations together, we get:
75p^2 = 3 * 5 * 5 * p * p
Now, let's find the powers of each prime factor:
- The power of 3 is 1 (since there is only one factor of 3).
- The power of 5 is 2 (since there are two factors of 5).
- The power of p is 2 (since there are two factors of p).
To determine the number of distinct positive factors, we need to consider the exponents of the prime factors. We add 1 to each exponent and take the product of these incremented values:
(1 + 1) * (2 + 1) * (2 + 1) = 2 * 3 * 3 = 18
Therefore, the minimum possible value of f(75p^2) is 18.