Difference between revisions of "2004 AMC 12B Problems/Problem 25"
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Revision as of 18:59, 3 July 2013
Problem
Given that 
is a 
-digit number whose first digit is 
, how many elements of the set 
have a first digit of 
?
Solution
Given 
digits, there must be a power of 
with digits such that the first digit is . Thus 
contains 
elements with a first digit of . For each number in the form of 
such that its first digit is , then 
must either have a first digit of or 
, and 
must have a first digit of 
. Thus there are also numbers with first digit 
and numbers with first digit 
. By using complementary counting, there are 
elements of with a first digit of 
. Now, has a first of if and only if the first digit of 
is , so there are 
elements of with a first digit of .
See also
| 2004 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 24 |
Followed by Last problem |
| 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 | |
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