I fulfilled the conditions for n so why is my answer wrong?

In my opinion, this problem is a hard concept one, so using the choosing numbers strategies might not be ideal. For example, choose n=48. It is easy to check that this number also satisfies all of the conditions, so in this case, the answer would be E, right? If you choose n=24, you will notice that all conditions are met as well, so the correct answer would be C. So what I am trying to say is, algebra is the better approach here.

Also, recall that **when choosing numbers, it is wise to choose more than one number**, so for example, if you choose n=72 as you did, you know that 12, 24, 36 all work, so we could discard answer choices A and E. The next step would be to choose a different number, for example the ones that I already enlisted. So, let n=48. In this case, notice that 36 doesn’t work anymore, so we can discard this answer choice; we are left with B, C. Now let n=12, and notice that B would work. What do all these three numbers have in common? That their greatest common factor, the concept that this question tests, is 12. Since the question asks for any number n such that all conditions are met, and given that we tested 3 numbers, we can say with some confidence that 12 is the largest positive integer that will divide n.