Difference between revisions of "2015 AMC 10A Problems/Problem 18"
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[[Category: Introductory Number Theory Problems]] | [[Category: Introductory Number Theory Problems]] |
Revision as of 01:18, 7 May 2020
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.) Therefore, our answer is
See Also
Video Solution: https://www.youtube.com/watch?v=2DVSkWu_H1g
2015 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 17 |
Followed by Problem 19 | |
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All AMC 10 Problems and Solutions |
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