Let us suppose that I have $n$ students in my class, and I break them up into $k$ groups per week. Let's also suppose that I want to repeat this each week, except that I don't want any student to work with any other student more than once.
For how many weeks can this continue?
This is an extension of some of the classic tournament-scheduling questions, and it wouldn't surprise me if there were an equally elementary approach. But I haven't found it yet. And since I haven't found it yet, I haven't come across the divisibility relationships that would facilitate a 'nice' answer, other than $k \mid n$.