Determine $\sigma (500)$.
We use the multiplicativity of the sigma function,
$$ \gcd(a,b)=1 \quad \implies \quad \sigma(a \times b) = \sigma(a) \times \sigma(b) $$
and Proposition (4.35), where $p$ is prime and $k$ is a positive integer,
$$ \sigma(p^k) = \frac{p^{k+1} - 1}{p - 1} $$
Noting $\gcd(2^2, 5^3)=1$, we have
$$\begin{align} \sigma(500) & = \sigma(2^2) \times \sigma(5^3) \\ \\ & = \frac{2^3-1}{2-1} \times \frac{5^4-1}{5-1} \\ \\ & = 1092 \end{align} $$