The Coin Collection

A PuzzleCritic Original:

The Museum of Mathematical Mysteries houses a peculiar collection of coins, each shaped like a polygon, with the two largest proper factors of n inscribed on either face, where n is the number of edges the given coin possesses. For example, there is a hexagonal coin on which the numbers 2 and 3 are inscribed. The museum’s curator examines one of the coins and sees the number 15 inscribed on one face. Determine all numbers that might appear on the other face.