AMC 12 2013b 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 \textbf{(B)}\ 10 \textbf{(C)}\ 15 \textbf{(D)}\ 20 \textbf{(E)}\ 25