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.
We are being asked to find all positive integers whose second or third largest factor is 15.
Though we might be able to make a few sensible guesses and stumble upon some correct answers, in order to be sure we have found them all, we need a logical argument.
Here a useful observation to make is that factors of numbers occur in natural pairs. For example, 20 can be written as 20 x 1, 10 x 2, or 5 x 4. The largest factor (20), is paired with the smallest factor (1), the second largest factor (10) is paired with the second smallest (2) and the third largest (5) is paired with the third smallest (4).
Let n be the number of edges the given coin possesses. Since 15 is either the second or third largest factor, n must be divisible by 1, 3, 5 and 15.
If 15 is the second largest factor of n, then we can write n=15d, where d is the second smallest factor of n. Given that 3 divides n, it follows that d = 2 or 3. This yields n = 30 or 45. In both cases , the number on the other side of the coin is the third largest factor of n, and we obtain solutions of 10 and 9 respectively.
If 15 is the third largest factor of n, then we can write n = 15e, where this time e is the third smallest factor of n. We know from the previous case that e cannot equal 2 or 3. Further, if e = 4, then n is divisible by 1, 2, 3 and 4, in which case e is no longer the third smallest factor, a contradiction. Since 5 must divide n, this leaves e = 5 as the only possibility. In this case, n = 75 and the number on the other face of the coin will be the second largest factor of n, namely 25.
Thus the complete set of solutions is 9, 10, 25.