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? 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?

Answer: So you have n students, where n>13, and m classrooms, where 3>m>13.

the question asked is: 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?

The only situation where it will be possible is when you take the total number of students, divide it by the number of classrooms and the result is a whole number (because here we are working with students, you can't have a 2/5 of a student, for example)

So n/m must be a natural number.

So now suppose that n is prime, this is : n only can be divided by itself, an example of a prime number is 17.

so if you have n = 17 students, there is no m that divides 17 into a whole number, then in this case, you can't assign the same number of students to each classroom.

And because we find a counterexample, so it is not possible for every n and m, so the statement is false. (independent of the fact that you actually could do this for some m and n given, the important thing here is that you can't do it for every combination of m and n)