Difference between revisions of "2015 AMC 10A Problems/Problem 18"
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Revision as of 09:47, 20 May 2015
Problem 18
Hexadecimal (base-16) numbers are written using numeric digits 
through 
as well as the letters 
through 
to represent 
through 
. Among the first 
positive integers, there are 
whose hexadecimal representation contains only numeric digits. What is the sum of the digits of ?
Solution
Notice that is 
in hexadecimal. We will proceed by constructing numbers that consist of only numeric digits in hexadecimal.
The first digit could be 
or 
and the second two could be any digit 
, giving 
combinations. However, this includes 
so this number must be diminished by 
Therefore, there are 
valid corresponding to those positive integers less than that consist of only numeric digits. (Notice that 
in hexadecimal.) Finally, our answer is 
See Also
| 2015 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 17 |
Followed by Problem 19 | |
| 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 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
