Data Sufficiency

Hello. Ive been reading this problem over and over again, but I think my confusion must come from the wording of each of the statements rather than the math. From the following problem:





I understand that a group of students will be divided into m groups of size n. Thus, n students per classroom. Total students nxm.

Additionally, the inequality implies that there is a number of classrooms m, between 4 and 12 inclusive. Each group of n students can be equal or greater than 14.

The goal is to determine if n can be divided into m without a remainder, given the restrictions, so each class has the same number of students without anyone being left out.

Statement (1)
3n/m yields an integer so one should check if it satisfies the restrictions for n and m.

N 3N | M
14 42 | 4
15 45 | 5
16 48 | 6
17 51 | 7
18 54 | 8
19 57 | 9
20 60 | 10
.... ... | 11
| 12

Now, from the 3N column, every number is divisible by some possible value of M except for 51 (3 x 17). This leads me to accept statement (1) as sufficient, given that it allows to answer affirmatively to the question "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?". Yes, they can be assigned in various combinations of N and M satisfying the restrictions.

I have gone through the explanation on the OG, but i cant seem to grasp why it is decided that the statement is not sufficient.

Im suspecting that I must be misinterpreting the wording in either the question and/or each of the statements.

I would appreciate if someone can provide their take on this one. Thanks!

Statement 1)

3N/M = Integer but
Case1- M could be a number made up of 3 and some factor of N e.g. N=32 and M=2x3=6
but N is NOT divisible by M

Case2- M could be a factor of N e.g. N=32 and M=8
but N is divisible by M

Insufficient

Statement 2)
13N/M = Integer
Since M has to be lesser than 13 (viz.a prime number) therefore M will have to be a factor of N only to result in an Integer when 13N is divided by M

Sufficient


Answer : B

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) It is possible to assign each of 3n students to one of m classrooms so that each classroom
has the same number of students assigned to it.

(2) It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students assigned to it.

To assign the same number of students to each classroom, the number of students (n) must be divisible by the number of classrooms (m).

Question rephrased: Is n/m an integer?

Statement 1: It is possible to assign each of 3n students to one of m classrooms so that each classroom
has the same number of students assigned to it.
In other words, the number of students (3n) is divisible by the number of classrooms (m).
Implication:
(3n)/m = 3(n/m) = integer.

Case 1: n/m = integer
It's possible that n=16 and m=4, with the result that n/m = 16/4 = 4.
Here, 3(n/m) = 3(16/4) = 12.

Case 2: n/m = k/3, where k is not a multiple of 3
In this case, since n/m = k/3, m must be a multiple of 3.
It's possible that n=14 and m=6, with the result that n/m = 14/6 = 7/3.
Here, 3(n/m) = 3(14/6) = 7.

Since n/m is an integer in Case 1 but not in Case 2, INSUFFICIENT.

Statement 2: It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students assigned to it.
In other words, the number of students (13n) is divisible by the number of classrooms (m).
Implication:
(13n)/m = 13(n/m) = integer.

Case 3: n/m = integer
It's possible that n=16 and m=4, since 16/4 = 4.
Here, 13(n/m) = 13(16/4) = 52.

Case 4: n/m = k/13, where k is not a multiple of 13
In this case, since n/m = k/13, m must be a multiple of 13.
Not possible, since 3 < m < 13.

Since only Case 3 is possible, n/m = integer.
SUFFICIENT.

The correct answer is B.

Hi All-

I'm having a tough time wrapping my head around the logical conclusion you draw in case 4 in both of your solutions. I believe there is an element of integer properties at play here I'm unaware of?

The given information tells us that 3 < m < 13 < n. Since m is between 3 and 13, there's no way that 13/m can be an integer. In fact, we can't even reduce the 13/m to simpler terms. How do we jump between these two statements? From this, we can conclude that n/m must be an integer.

Thanks in advance!

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