2021 Fall AMC 12B Problem 12

For n a positive integer, let f(n) be the quotient obtained when the sum of all positive divisors of n is divided by n. For example, f(14)=(1+2+7+14)\div 14=\frac{12}{7} What is f(768)-f(384)?

\textbf{(A)}\ \frac{1}{768} \qquad\textbf{(B)}\ \frac{1}{192} \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ \frac{4}{3} \qquad\textbf{(E)}\ \frac{8}{3}

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