This problem gave me a great deal of trouble when I encountered it. I figured out quickly enough that the answer to the maximum case was to make the largest team possible and put as many participants in groups of size 1 in order to fulfill the requirement of having m groups. I also figured out that the solution for the minimum case was to put students in as "uniform" of groups as possible. It was figuring out how to put students in uniform groups which was the challenge.
In hindsight, the main difficulty was that I was lacking the interpretation of n % m necessary for this problem. The perspective that has worked 99% of the time for me was to view the quanity n % m as the number r such that n = k*m + r, where 0 <= r < m. Intutively, this can be thought of as repeatedly adding m until it gets within m of n and then calling the remainder r.
Another way to view this procedure is that we are making as many groups of size m out of n as we possibly can. Unfortunately this problem calls for making exactly m groups, which are almost certainly not of size m.
As a review, n / m expresses the answer to each of the two questions:
- split n pieces into m groups. how many are in each group?
- split n pieces into groups of size m. how many groups are there?