Source: IMS GMAT

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.
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.

First, we rephrase the question. We are asked whether it is possible to divide up n students into m equal groups. In other words, we are being asked "Is n divisible by m?".

We can analyze this question further in terms of prime factors. A number n is divisible by a number m if, and only if, the prime box for n contains all of the numbers in the prime box for m. Thus 60 is divisible by 12 because the prime box for 60 [2,2,3,5] contains all of the members of the prime box for 12 [2,2,3]. And 70 is not divisible by 4 because the prime box for 70 [2,5,7] does not contain all of the members of the prime box for 4 [2,2] - the prime box for 4 has an extra 2 that isn't in the prime box for 70.

So we are really being asked: "Does the prime box for n contain all of the members of the prime box for m?"

(1) INSUFFICIENT. If 3n is divisible by m, the prime box for 3n contains all of the prime factors of m. This doesn't imply that n is divisible by m, however. If m is a multiple of 3, such as 6, any number that is divisible by m must have a 3 in its prime box. 3n clearly has a 3 in its prime box, but we cannot be certain that n does. (Imagine, for instance that n=20 and m=6. 3n would be divisible by 6, but n would not be.]

(2) SUFFICIENT. If 13n is divisible by m, the prime box for 13n contains all of the prime factors of m. Since m is less than 13, we know that 13 cannot be a prime factor of m. Therefore, the prime box for n, which is just the prime box for 13n without the extra 13, will also have all of the prime factors of m. Thus n itself must be divisible by m.

The answer is B.

(For more on prime boxes, see the ManhattanGMAT "Number Properties" study guide.)

Great Explanation Jad.

GMAT 2007