2014 Pascal Problem 21

Twenty-four identical 1 \times 1 squares form a 4 \times 6 rectangle, as shown. A *lattice point* is a point where a horizontal grid line intersects a vertical grid line. A diagonal of this rectangle passes through the three lattice points P, Q and R. When a 30 \times 45 rectangle is constructed using identical 1 \times 1 squares, how many lattice points will a diagonal of this rectangle pass through?

\textbf{(A)}\ 19\quad \textbf{(B)}\ 16\quad \textbf{(C)}\ 15\quad \textbf{(D)}\ 18\quad \textbf{(E)}\ 12

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.