2025 AMC 8 Problem 22

A classroom has a row of 35 coat hooks. Paulina likes coats to be equally spaced, so that there is the same number of empty hooks before the first coat, after the last coat, and between every coat and the next one. Suppose there is at least 1 coat and at least 1 empty hook. How many different numbers of coats can satisfy Paulina’s pattern?

\textbf{(A)} 2 \qquad \textbf{(B)} 4 \qquad \textbf{(C)} 5 \qquad \textbf{(D)} 7 \qquad \textbf{(E)} 9

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