2013 AMC 12B Problem 23

Bernardo chooses a three-digit positive integer N and writes both its base-5 and base-6 representations on a blackboard. Later LeRoy sees the two numbers Bernardo has written. Treating the two numbers as base-10 integers, he adds them to obtain an integer S. For example, if N=749, Bernardo writes the numbers 10444 and 3245, and LeRoy obtains the sum S=13,689. For how many choices of N are the two rightmost digits of S, in order, the same as those of 2N?

\textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 25

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