2013 AMC 10B Problems/Problem 25
- The following problem is from both the 2013 AMC 12B #23 and 2013 AMC 10B #25, so both problems redirect to this page.
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 and , and LeRoy obtains the sum . For how many choices of are the two rightmost digits of , in order, the same as those of ?
First, we can examine the units digits of the number base 5 and base 6 and eliminate some possibilities.
Substituting these equations into the question and setting the units digits of 2N and S equal to each other, it can be seen that , and , so , , ,
Therefore, can be written as and can be written as
Keep in mind that can be , five choices; Also, we have already found which digits of will add up into the units digits of .
Now, examine the tens digit, by using and to find the tens digit (units digits can be disregarded because will always work) Then we see that and take it and to find the last two digits in the base and representation. Both of those must add up to
Now, since will always work if works, then we can treat as a units digit instead of a tens digit in the respective bases and decrease the mods so that is now the units digit.
Say that (m is between 0-6, n is 0-4 because of constraints on x) Then
and this simplifies to
From inspection, when
This gives you choices for , and choices for , so the answer is
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