Question 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 $10{,}444$ and $3{,}245,$ 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 $2N?$