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