Euclid's formula/Perfect Numbers #2

[imprank6]

Homework Statement

Prove that the sum of the reciprocals of all divisors of a perfect number is equal to 2.

Homework Equations

Euclid's formula for perfect #'s

The Attempt at a Solution

I am so lost.

 

[VeeEight]

Let d₁, d₂,..., d_k be a (ordered) list of divisors of perfect number n where d₁ = 1, d_k = n. It is clear that the sum of all these divisors is 2n.

Observe that d₁d_k = n implies d₁ = n/d_k, which in turn implies that 1/d₁ = d_k/n.

Do this same thing for d₂d_k-1 = n implies... etc
Now you have each reciprocal written as d/n where d is a divisor of n, so add them up.