Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number "M" is exactly "2M"; therefore, the average of the divisors is "M"(2/τ("M")), where τ("M") denotes the number of divisors of "M". For any "M", τ("M") is odd if and only if "M" is a square number, for otherwise each divisor "d" of "M" can be paired with a different divisor "M"/"d". But, no perfect number can be a square: this follows from the known form of even perfect numbers and from the fact that odd perfect numbers (if they exist) must have a factor of the form "q"α where α ≡ 1 (mod 4). Therefore, for a perfect number "M", τ("M") is even and the average of the divisors is the product of "M" with the unit fraction 2/τ("M"); thus, "M" is a harmonic divisor number.