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?
Oh now I see why N and M must be odd! Otherwise, the polygon can be made irregular by making two opposite sides longer or shorter by the same amount.
I concur that (9,3) is the only answer.
Aha!
