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Difference between revisions of "2015 AMC 10A Problems/Problem 18"

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Notice that <math>1000</math> is <math>3E8</math> in hexadecimal.  We will proceed by constructing numbers that consist of only numeric digits in hexadecimal.
 
Notice that <math>1000</math> is <math>3E8</math> in hexadecimal.  We will proceed by constructing numbers that consist of only numeric digits in hexadecimal.
  
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The first digit could be <math>0,</math> <math>1,</math> <math>2,</math> or <math>3,</math> and the second two could be any digit <math>0 - 9</math>, giving <math>4 \cdot 10 \cdot 10 = 400</math> combinations.  However, this includes <math>000,</math> so this number must be diminished by <math>1.</math>  Therefore, there are <math>399</math> valid <math>n</math> corresponding to those <math>399</math> positive integers less than <math>1000</math> that consist of only numeric digits.  (Notice that <math>399 < 3E8</math> in hexadecimal.)  Therefore, our answer is <math>3 + 9 + 9 = \boxed{\textbf{(E) } 21}</math>
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The first digit could be <math>0,</math> <math>1,</math> <math>2,</math> or <math>3,</math> and the second two could be any digit <math>0 - 9</math>, giving <math>4 \cdot 10 \cdot 10 = 400</math> combinations.  However, this includes <math>000,</math> so this number must be diminished by <math>1.</math>  Therefore, there are <math>399</math> valid <math>n</math> corresponding to those <math>399</math> positive integers less than <math>1000</math> that consist of only numeric digits.  (Notice that <math>399</math> is the least hexadecimal number using only decimal digits before <math>3E8</math>.)  Therefore, our answer is <math>3 + 9 + 9 = \boxed{\textbf{(E) } 21}</math>
  
 
==See Also==
 
==See Also==

Revision as of 21:49, 4 July 2020

Problem 18

Hexadecimal (base-16) numbers are written using numeric digits $0$ through $9$ as well as the letters $A$ through $F$ to represent $10$ through $15$. Among the first $1000$ positive integers, there are $n$ whose hexadecimal representation contains only numeric digits. What is the sum of the digits of $n$?

$\textbf{(A) }17\qquad\textbf{(B) }18\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

Solution

Notice that $1000$ is $3E8$ in hexadecimal. We will proceed by constructing numbers that consist of only numeric digits in hexadecimal.

The first digit could be $0,$ $1,$ $2,$ or $3,$ and the second two could be any digit $0 - 9$, giving $4 \cdot 10 \cdot 10 = 400$ combinations. However, this includes $000,$ so this number must be diminished by $1.$ Therefore, there are $399$ valid $n$ corresponding to those $399$ positive integers less than $1000$ that consist of only numeric digits. (Notice that $399$ is the least hexadecimal number using only decimal digits before $3E8$.) Therefore, our answer is $3 + 9 + 9 = \boxed{\textbf{(E) } 21}$

See Also

Video Solution: https://www.youtube.com/watch?v=2DVSkWu_H1g

2015 AMC 10A (Problems • Answer Key • Resources)
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Problem 17 		Followed by
Problem 19
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