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?

As what is sought are polygons that have angles of equal integral degrees that, inscribed in a circle, must have equal side lengths, the number of sides must be odd.
The odd regular n-gons of integral internal angles are 3, 5, 9, 15 and 45. Of these, the only polygon with an odd number of divisors is for the nonagon (9-gon): 3 {1,3,9}, thus the answer must be the 9 and 3.
 

 

 

Edited on May 28, 2010, 2:04 pm

Posted by Dej Mar on 2010-05-28 13:26:20