Math Gold Medalist

Lor

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 $f(2023)$?

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

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   Solution