2007 AMC 12B Problems/Problem 21
Problem
The first positive integers are each written in base . How many of these base- representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
Solution
All numbers of six or less digits in base 3 have been written.
The form of each palindrome is as follows
1 digit -
2 digits -
3 digits -
4 digits -
5 digits -
6 digits -
Where are base 3 digits
Since , this gives a total of palindromes so far.
7 digits - , but not all of the numbers are less than
Case:
All of these numbers are less than giving more palindromes
Case: ,
All of these numbers are also small enough, giving more palindromes
Case: ,
It follows that , since any other would make the value too large. This leaves the number as . Checking each value of d, all of the three are small enough, so that gives more palindromes.
Summing our cases there are
See Also
2007 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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