Question

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.

Answer 1

The correct option is B Statement (2) alone is sufficient, but statement (1) alone is not sufficient.

Determine if n is divisible by m.

1. Given that 3n is divisible by m, then n is divisible by m if m = 9 and n = 27 (note that 3 < m < 13 < n, 3n = 81, and m = 9, so 3n is divisible by m) and n is not divisible by m if m = 9 and n = 30 (note that 3 < m < 13 < n, 3n = 90, and m = 9, so 3n is divisible by m); NOT sufficient.
2. Given that 13n is divisible by m, then 13n = qm for some integer q. Since 13 is a prime number that divides qm (because 13n = qm) and 13 does not divide m (because m < 13), it follows that 13 divides q. Therefore, is an integer, and since , then is an integer. Thus, n is divisible by m; SUFFICIENT.

• The correct answer is B; statement 2 alone is sufficient.