Answer: Hello!
first, we have m classrooms and n students, where 3<m<13<n
a) 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?
If this statement is true, then it should be true for all the m between 3 and 13, and all the n greater than 13.
If we want that every classroom has the same amount of students. then n/m should be an integer number.
Let's find a counterexample!
if n is 17, then n is a prime number. this means that only can be divided by itself and 1. Then m doesn't divide n, and we cant divide 17 students in any amount of classrooms. This is only possible if n is a multiple of m.
b) if now we have 3n students, again we need to find a counterexample where 3n is not divisible by m. this is only possible if 3n is a multiple of n.
suppose that n = 20, then 3n= 60, and we need to find a number m that cant divide 60 to prove that this is false.
then if m = 7, 60/7 = 8.5 is not an integer, then, in this case, we cant divide 3n students in m classrooms.
the only situation where you can divide 3n students into m classrooms is if 3n is a multiple of m
c) now we have 13n students and m classrooms again.
if n = 13, then 13n = 13*13.
now, 13 is also a prime number, then 13*13 can only be divided by 1, 13 and itself.
then there is no m that divides this number.
in this case, because there is no m that divides 13 (again, because 13 is prime) the only situation where you can divide the 13n students into m classrooms is where n is a multiple of m.
