fangtray wrote:A school administrator will assign each student in a group of n students to one of m classrooms. If 3 < m < 13 < n, is it possible to assign each of the n students to one of the m classrooms so that each classroom has the same number of students assigned to it?

1) 3n divisible by m
2) 13n divisible by m

on the explanation for 1) the book states...

given that 3n is divisible by m, then n is divisible by m if m = n = 9 and n is not divisible by m if m = 9 and n = 12. NOT Sufficient. How can n = 9 or 12 if the question states that n is > 13??

Hi fangtray!

Let me first agree with you wholeheartedly that the OG Explanations are often WORSE than simply not knowing (they can be ridiculously confusing and I believe they intentionally write them to be opaque)! The question stem does say that n>13, so the discussion of n=9 is just madness

As for an easier explanation for this problem, how about the following:

Question Stem:
The long drawn out story is really just asking if N is divisible by M given that M could be 4, 5, 6, 7, 8, 9, 10, 11 or 12. is N a multiple of one of those numbers??

Statement 1:
We know that 3N is divisible by M, but what about just N? Well, that just depends on what impact the 3 is having (if any). This can be a bit tough to see so I'd actually suggest skipping to Statement (2) and coming back...

Statement 2:
We know that 13N is divisible by M, but what about just N? Well, again, this will depend on the impact of the 13. What I mean is - does that 13 help make N divisible by M? Well, let's look at the possible values of M again...3, 4, 5, 6, 7, 8, 9, 10, 11, 12... is that 13 divisible by any of those numbers? Well, because 13 is a prime (no factors besides itself and 1), it is NOT divisible by any of the possible values of M. That means that for 13N to be divisible by M, then the N has to do all the work. Put a different way, all of the pieces of M have to be found in the N. That means that N must be divisible by M. [spoiler]Sufficient.[/spoiler]

Statement 1 Revisited:
Okay, so let's return to the idea of 3N divisible by M. It might make a bit more sense now when I ask if 3 is doing any of the work to make 3N divisible by M. For example, if M is equal to something like 6, 9 or 12, then it might be the case that the 3 from 3N is actually helping.

3N/6 = N/2 --> so we only know for sure that N is divisible by 2 (but not our M=6)
3N/9 = N/3 --> so we only know for sure that N is divisible by 3 (but not our M=9)
3N/12 = N/4 --> so we only know for sure that N is divisible by 4 (but not our M=12)

But there are some scenarios where the 3 from 3N wouldn't be helping:

3N/4 = 3N/4 --> we can't reduce, so if this number is an integer, then the N must be divisible by our m=4

So Statement 1 is actually [spoiler]NOT sufficient![/spoiler]

That gives us the correct answer: [spoiler]B.[/spoiler]