Difference between revisions of "2007 AMC 12B Problems/Problem 21"
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The first <math>2007</math> positive integers are each written in base <math>3</math>. How many of these base-<math>3</math> representations are palindromes? (A palindrome is a number that reads the same forward and backward.) | The first <math>2007</math> positive integers are each written in base <math>3</math>. How many of these base-<math>3</math> representations are palindromes? (A palindrome is a number that reads the same forward and backward.) | ||
− | <math>\mathrm {(A)} 100 | + | <math>\mathrm {(A)}\ 100 \qquad \mathrm {(B)}\ 101 \qquad \mathrm {(C)}\ 102 \qquad \mathrm {(D)}\ 103 \qquad \mathrm {(E)}\ 104</math> |
==Solution== | ==Solution== | ||
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==See Also== | ==See Also== | ||
{{AMC12 box|year=2007|ab=B|num-b=20|num-a=22}} | {{AMC12 box|year=2007|ab=B|num-b=20|num-a=22}} | ||
+ | [[Category:Introductory Combinatorics Problems]] |
Revision as of 22:09, 21 June 2009
Problem 21
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 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |