2022 Pascal Problem 20

A pizza is cut into 10 pieces. Two of the pieces are each \frac{1}{24} of the whole pizza, four are each \frac{1}{12}, two are each \frac{1}{8}, and two are each \frac{1}{6}. A group of n friends share the pizza by distributing all of these pieces. They do not cut any of these pieces. Each of the n friends receives, in total, an equal fraction of the whole pizza. The sum of the values of n with 2 \le n \le 10 for which this is *not* possible is

\textbf{(A)}\ 31\quad \textbf{(B)}\ 35\quad \textbf{(C)}\ 40\quad \textbf{(D)}\ 39\quad \textbf{(E)}\ 36

If there are no answer choices shown, enter a numerical answer.

Full credit to this problem is given to the CEMC, you may view all Pascal contests here.