Problem 169 from Quantitative Review 2nd Ed

dghosh2602 wrote:I was hoping to get some clarification on Problem 169 from Quantitative Review 2nd Ed:

Q: If n is a positive integer and n^2 is divisible by 72, then the largest positive integer that must divide n is:
A 6, B 12, C 24, D 36, E 48

n^2 is divisible by 72, but it must also be greater than 72. If n is an integer, then n^2 must be a perfect square. The factorization of 72 is (8)(9), so if it is multiplied by 2, it will be (2)(8)(9) = (16)(9) = 144, a perfect square. So n^2 must be at least 144 or a multiple of 144, which means that n must be 12 or a multiple of 12.

I know that Quantitative Review also has 12 as the answer, but I had a question: Since n must be 12 or a multiple of 12, why is it that 48 isn't a solution since its a multiple of 12 and 48 divides 48 and is also the greatest number amongst the solutions, especially because the question does not state 'largest integer other than n that divides n'? What is the concept that I am not getting?

Please help.

debmalya_dutta wrote:n^2 = 72 K as provided
so for (2*2)*(3*3) *2 * K to be a perfect square of n , K = 2 * {square of some number}

so n^2 = 72 K = (2*2)*(3*3) *2 * 2 * {square of some number } - lets call square of some number = a^2
n=square root of [ (2*2)*(3*3) *2 * 2 * {a^2}
n = 2 * 3 * 2 * a
n=12a
So the largest number that always divides n is 12