Posted from my mobile device

please explain this once again i didnt understand why 6

-----ASIDE---------------------
A lot of integer property questions can be solved using prime factorization.
For questions involving divisibility, divisors, factors and multiples, we can say:
If N is divisible by k, then k is "hiding" within the prime factorization of N

Consider these examples:
24 is divisible by 3 because 24 = (2)(2)(2)(3)
Likewise, 70 is divisible by 5 because 70 = (2)(5)(7)
And 112 is divisible by 8 because 112 = (2)(2)(2)(2)(7)
And 630 is divisible by 15 because 630 = (2)(3)(3)(5)(7)
-----ONTO THE QUESTION!---------------------

If n³ is divisible by 24, then we can say that n³= 24k for some positive integer k

Since 24 = (2)(2)(2)(3), we can write: n³= (2)(2)(2)(3)k for some positive integer k

So, what does this tell us about n?
Since there's an 8 hiding in the prime factorization of n³, we can conclude that 8 is a divisor of n³, which means 2 must be a divisor of n
It also tells us that 3 must be a divisor of n
These are the only two guaranteed conclusions we can draw about the divisors of n

If 2 and 3 must be divisors of n, then we can be certain that and is divisible by 6

Answer: C

ohhhh i got it now thank you sir

I have follow the process below-
n^3/24 so we can say that n^3 must have at least 3 two's and 1 three. As n^3= 2^3 so n=2^1 or 2 and again as n^3= 3 or 3^1, thus n will be 3^3. so the n could be 2X3X3X3= 54. The factors of 54 are- 1, 2, 3, 6, 9, 18, 27 etc. So from the answer choices 6 is the largest one. Is it a correct way?

N should contain atleast one 2 and one 3.so N Can values [6,12,18,...].if we take 6 or 12 then both options 6 and 12 are correct but if we take N as 18 only 6 is a factor of N but 12 is not.So 6 is the answer.

Posted from my mobile device

n^3 = 24k

n = 2*(3^1/3)*(k^1/3)

n needs to be a positive integer according to the options

that means (3k)^1/3 needs to be a positive integer and k = 9 satisfies this resulting n = 6

k = 3^5 is the next best number resulting n = 18

and it goes on in k = 3^(2+m) where 0 <= m <= infinite

you can go further like this but im gonna stop here

both 18 and 6 have HCF of 6

option C it is

the solution by GreenlightTestprep is the standard approach and is a must way to approach this problem

my solution isnt quite....standard but it is interesting to say the least