2007 Cayley Problem 22

Five positive integers are listed in increasing order. The difference between any two consecutive numbers in the list is three. The fifth number is a multiple of the first number. How many different such lists of five integers are there?

\textbf{(A)}\ 3\quad \textbf{(B)}\ 4\quad \textbf{(C)}\ 5\quad \textbf{(D)}\ 6\quad \textbf{(E)}\ 7

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 Cayley contests here.