Saw this question in the second arithmetic class. Didn’t understand why 72 and 24 weren’t chosen. If we do 72 squared it equals 5,184, which is divisable by both 108 and 24. And then getting divisors of 72, we get all of the options. Can someone illustrate why, this options shouldn’t be selected.
S is a set of all positive multiples of 36. We want to select numbers that are going to be factors for all elements in S.
We know that 36 is an element of S, but obviously 24 and 72 do not divide it (i.e., they are not factors of 36). If a number fails to be a factor of just one element in S, it can’t possibly be a factor of all elements in S.
Thanks for the explanation. Still a bit confused on why S is a set of multiples of 36 and not multiples of 72. 72 also applies for the rule of n squared (72 squared = 5,184) being a multiple of 24 and 108. All numbers on the answer choice are then divisible by 72.
Is 36^2 a multiple of both 24 and 108? If so, then 36 must be in S. All multiples of 72 are indeed contained in S, but there are more elements besides that as well, such as 36, 36(3), 36(5), and so on.
Thanks again for your reply, but I’m not fully understanding your point. Another way to understand what I’m trying to explain is finding the LCM of 24 and 108. This is 216, which can also be seen as 2^3 * 3^2. This shows that n can be any value which basically can factor this. 36 is one example, but 72 contains this factors. This means that n can be 36 OR 72. As the answer states to select ALL such integers, imo, the correct answer should be A,B,C,D
Correct so n^2 can be any multiple of 216, right?
Not sure what this is supposed to mean, but maybe this is the point of confusion, since it does seem incorrect.
So you do seem to agree that S contains at bare minimum 36 and 72.
S = \{36,72,… \}
You have to pick answers such that, for all elements in S, the answer is a factor to it. B and D are certainly factors of 72, but not of 36, so you can’t pick those numbers. The correct answers are the two numbers that are factors of both 36 and 72.