AMC 12A 2023 (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)

\text{(A)}\;-1536

\text{(B)}\;96

\text{(C)}\;108

\text{(D)}\;116

\text{(E)}\;144