2023 AMC 12A 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 \qquad \textbf{(B)} 96 \qquad \textbf{(C)} 108 \qquad \textbf{(D)} 116 \qquad \textbf{(E)} 144

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.