Concept: This is a very good question which tests your knowledge of Number properties. Let us break down the question statement and the question stem:

Solution:

There are N students and M classrooms. Each of these N students will be assigned to sit in one of the M classrooms.

3<M<13<N – the minimum number of rooms = 4 and maximum number of rooms = 12 and the minimum number of students = 14. So, 4≤M≤12 and N≥14.

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 – Is N divisible by M could be the simplified version of the question statement.

From statement I alone, we can infer that 3N is divisible by M. This is not sufficient to find out if N is divisible by M.

If N = 15 and M = 5, 3N = 45 is divisible by M = 5; also, N =15 divisible by M = 5, so we answer the question with a YES.
If N = 21 and M = 9, 3N = 62 is divisible by M = 9; but N = 21 is not divisible by M = 9, so we answer the question with a NO.

Statement I alone is insufficient to obtain a definite YES or NO. Answer options A and D can be eliminated. Possible answer options are B, C or E.

From statement II alone, we can infer that 13N is divisible by M. This is where we need to be more careful and just not replicate what we did with statement I.

Note that 13 is a prime number and is only divisible only by 1 and 13. Therefore, 13 is not divisible by M since M is neither of these two numbers.
Therefore, the only way in which 13N is perfectly divisible by M is because N is divisible by M.

Statement II alone is sufficient to answer the question with a YES. Answer options C and E can be eliminated.

 The correct answer option is B.