There is a number N such that if you inscribe an N-gon in a circle, with all its angles an equal number of integral degrees, that N-gon will necessarily be a regular N-gon, with equal side lengths.
Also, if M is the number of possible divisors of N, including 1 and N itself, then the number M also meets the same criteria: If you inscribe an M-gon in a circle, with all its angles an equal number of integral degrees, that M-gon will necessarily be a regular M-gon, with equal side lengths.
What are N and M?
I don't see any reason that N and M need to be odd. All we need is N and M to both divide 360 evenly, and be greater than 2.
In addition to (9,3), I think that (6,4), (10,4), (12,6), and (18,6) all do it. Also (20,6), (24,8), (36,9), (90,12), (180,18), (360,24), and possibly others that I have missed.
Am I misunderstanding the problem? Have I miscalculated (all done by hand)
More answers? (SPOILER?)
