A School Administrator will Assign each Student in a Group GMAT Data Sufficiency

(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.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

“A School Administrator will Assign each Student in a Group of N Students to One of M Classrooms GMAT Data Sufficiency”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiencycomprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation

Approach Solution1

It is asked Can all n students be placed in one of the m classrooms so that each one has the same amount of students assigned to it?

Let us rephrase the question
The number of students (n) must be divisible by the number of classes in order to be able to put the same number of pupils in each classroom (m). That is to say, n/m has to be an integer.

Target question is now = Is n/m an integer?

Statement 1 says
It is possible to assign each of 3n students to one of m classrooms so that each classroom has the same number of students to it.

This statement demonstrates that the number of students (3n) is divisible by the number of classroom(m). In other words, 3n/m is an integer.

Does this imply that m/n is a integer? No.
Think about these instances that conflict.
Case A: If n = 20 and m = 4, then n/m is an integer.
Case B: In this scenario, n/m is not an integer because m = 6 and n = 20.
Because we are unable to definitively respond to the REPHRASED target question, statement 1 is insufficient.

Statement 2 says
It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students to it

This statement demonstrates that the number of students (13n) is divisible by the number of classrooms (m). Which means, 13n/m is an integer.

According to the information provided, 3 < m < 13 < n.
There is no possibility that 13/m may be an integer because m is an integer between 3 and 13.

We can infer from this that n/m must be an integer.

Since, The REPHRASED target question can be answered with certainty, hence statement 2 is sufficient.

Correct answer: B

Approach Solution 2

There is another approach to answering this question

We want to know, Do we have the ability to divide N and M evenly?
We already know that 3 < M < 13 < N.

Statement 1 says
It is possible to assign each of 3n students to one of m classrooms so that each classroom has the same number of students to it.

This demonstrates that 3N can be divided by M. Since M is higher than 3, it may be 4/5/6/7, etc.

Now take note that M might be 5, in which case it needs to be a multiple of N. The only element of N that needs to be a multiple of 2 is if M is 6 (a multiple of 3). In this situation, N might not be divisible by 6.
So,
M = 5, N = 20 is possible
M = 6, N = 14 is also possible

Hence this statement 1 alone is insufficient.

Statement 2 says
It is possible to assign each of 13n students to one of m classrooms so that each classroom has the same number of students to it

As a result, we know that 13N is divisible by M. M is less than 13, hence neither M nor any factor containing 13 may be 13. Accordingly, N must be divisible by M if 13N is to be divisible by M.
For instance, M must be a factor of 14 if M (which is less than 13) can divide 13*14.

Hence this statement 2 alone is sufficient.

Correct answer: B.

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