Prove that for all positive integersaandb, σ(ab) ≤ σ (a) σ (b).

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- Aug 20th 2012, 09:03 PMmusngiburgerProving on positive integers?
Prove that for all positive integers

*a*and*b*, σ(ab) ≤ σ (a) σ (b). - Aug 20th 2012, 11:21 PMVlasevRe: Proving on positive integers?
Is this the sum of divisors function?

- Aug 21st 2012, 08:31 AMrichard1234Re: Proving on positive integers?
The sum of divisors function $\displaystyle \sigma$ is multiplicative for relatively prime integers. We will consider any common prime factor (p) of a and b. Suppose that $\displaystyle p^k$ completely divides $\displaystyle a$ and $\displaystyle p^m$ completely divides $\displaystyle b$.

You want to prove that

$\displaystyle \sigma(p^{k+m}) \le \sigma (p^k) \sigma (p^m)$

However, finding the sum of divisors of a prime power is easy (just a geometric series).

$\displaystyle 1 + p + \dots + p^{k+m} \le (1 + p + \dots p^k)(1 + p + \dots p^m)$

And you can go on from there.