2015 AMC 10A Problems/Problem 18
Contents
Problem
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 when converted to 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 is the least hexadecimal number using only decimal digits before .) Therefore, our answer is
Reasoning for the 400 Combinations
This is because we don't want more than for each digit since that would lead to a letter digit(A, B, C, D, E, or F.) Thus we have Then continue as follows.
~Clarification by mathboy282
Solution 2 (Casework)
First, we set a bound by writing in base-. . Therefore, we are considering numbers with a maximum of digits, and a maximum of in the ths-place (the first place in a -digit number).
Case : -digit numbers: There are evidently numbers that fit this category.
Case : -digit numbers: There are numbers that fit this category.
Case : -digit numbers: There are numbers that fit this category
Adding these up, we get numbers. ~sosiaops
Solution 3 (Quick)
We can quickly see that in hexadecimal = = 1024. If we go down to 399 in hexadecimal, we have which is , which is obviously less than 1000. Therefore, the answer is =
~Arcticturn
Video Solutions
https://youtu.be/ZhAZ1oPe5Ds?t=4596
https://www.youtube.com/watch?v=2DVSkWu_H1g
See Also
2015 AMC 10A (Problems • Answer Key • Resources) | ||
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Followed by Problem 19 | |
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