2013 AMC 12B Problems/Problem 23

Problem

Bernardo chooses a three-digit positive integer 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 . For example, if , Bernardo writes the numbers 10,444 and 3,245, and LeRoy obtains the sum . For how many choices of are the two rightmost digits of , in order, the same as those of ?

$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 10\qquad\textbf{(C)}\ 15\qquad\textbf{(D}}\ 20\qquad\textbf{(E)}\ 25$ (Error compiling LaTeX. Unknown error_msg)