Hi, I need help with this question, my thinking was that N should atleast have 2^2 * 3^2 so that N^2 will always a multiple of both 24 & 108 (hence chose C & D). Where did I go wrong?

Q. Let S be the set of all positive integers n such that n 2 is a multiple of both
24 and 108. Which of the following integers are divisors of every integer n
in S ?
Indicate all such integers.

Your analysis is correct that N should at least have 2^2 * 3^2. Next, you need to select the answer options which are divisors of N, that is, the numbers which go completely into N (N needs to be greater or at most equal to these numbers).

Looking at option D, the prime factorisation of 72 is 2 × 2 × 2 × 3 × 3= 2^3 * 3^2

This is not a divisor of N as N only has two 2s, not three 2s. For the answer option to be a divisor, its prime factors should be completely subsumed in the prime factors of N. Can you identify the correct options using this information?