Problem: The sum of positive integers less than $n$ and relatively prime to $n$ is equal to $1008n$, then find the value of $\varphi (n)$, where $\varphi (n)$ is the number of positive integers less than $n$ and relatively prime to $n$.
Solution: We recall the following result.
Therefore, using the above theorem, we have \begin{align*} \frac{n \varphi (n)}{2} = 1008n & \implies \frac{ \varphi (n)}{2} = 1008 \\ & \implies \varphi (n) = 2016. \end{align*} Hence, \[ \textcolor{blue}{\boxed{\varphi (n) = 2016}} \]