Prove that for all positive integers a and b, σ(ab) ≤ σ (a) σ (b).
The sum of divisors function is multiplicative for relatively prime integers. We will consider any common prime factor (p) of a and b. Suppose that completely divides and completely divides .
You want to prove that
However, finding the sum of divisors of a prime power is easy (just a geometric series).
And you can go on from there.