2020 Pascal Problem 25

We call the pair (m, n) of positive integers a *happy pair* if the greatest common divisor of m and n is a perfect square. For example, (20, 24) is a happy pair because the greatest common divisor of 20 and 24 is 4. Suppose that k is a positive integer such that (205\,800, 35k) is a happy pair. The number of possible values of k with k \le 2940 is

\textbf{(A)}\ 36\quad \textbf{(B)}\ 28\quad \textbf{(C)}\ 24\quad \textbf{(D)}\ 30\quad \textbf{(E)}\ 27

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.