AMC 12 2023a Problem 22

Let $f$ be the unique function defined on the positive integers such that$$\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1$$for all positive integers $n$, where the sum is taken over all positive divisors of $n$. What is $f(2023)$?

\textbf{(A)}~-1536 \textbf{(B)}~96 \textbf{(C)}~108 \textbf{(D)}~116 \textbf{(E)}~144