a similar problem was discused few days bck on this forum. let me try this again.
n^2 is divisible by 72, so n^2=72k (k is any inetegr- n^2 is a multiple of 72)
so n=sqrt(72k)=6*sqrt(2k)
the question asks for a +ve integer tht MUST divide n,->whtevr the value of n, it shud be divisible by this integer.
now n shud be an integer,so lets see for which values of k does n become an integer,
the least value of k for which n becomes an integer is k=2, then n =12
so 12 is the iteger which divides n always.

option B.

Thanks for the answer. Can someone expand on this?

To summarize..
- n is positive
- n^2 is divisible by 72
- what is the largest positive integer that must divide n?

1. If they are asking for the "largest positive integer that must divide n"; why would we use the smallest possible (integer) value of "k" to solve the question? For example, if we used "k=8" the answer would be (c)24.

Cheers,

important thgn to note here is the word MUST, whatevr the value of n, this integer MUST divide it.
if we take the example you mentioned, if k=8, n woild be 24.
if 24 is the largest possible integer we are looking for, how abt the below case?
k=2,n=12. remember 24 MUST divide n, but 24 doesnt divide 12.
so we are finding the min. value of n so tht if for tht value the integer divides n, rest all wud just be multiples of tht particular n.
largest positive integer, - both 6 and 12 would divide all values of n, the larger one would be 12.

hope this helps.

The words "divisibe by" are your clue that this is about factors.

if n^2 is divisible by 72 that means that n^2 must have all the prime factors 72

72 has 2,2,2,3,3 as it's prime factors.

since it is n^2 that has these we know that n has a 2 and a 3 (by pulling out the factor pairs) - but there is a still a 2 left and thus n must have another 2 in it (because the square is divisible and thus can't be a decimal). thus n must have 2, 2, and 3 as factors (can have many more but must have these).

Thus n is divisible by 12 - n could be divisible by the larger numbers but we don't know for sure a thus the answer is B.

Best of Luck