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Difference between revisions of "2005 AMC 8 Problems/Problem 18"

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<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77</math>
 
<math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77</math>
  
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==Video Solution==
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==Solution==
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Let <math>k</math> be any positive integer so that <math>13k</math> is a multiple of <math>13</math>. For the smallest three-digit number, <math>13k>100</math> and <math>k>\frac{100}{13} \approx 7.7</math>. For the greatest three-digit number, <math>13k<999</math> and <math>k<\frac{999}{13} \approx 76.8</math>. The number <math>k</math> can range from <math>8</math> to <math>76</math> so there are <math>\boxed{\textbf{(C)}\ 69}</math> three-digit numbers.
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==Video Solution by OmegaLearn==
 
https://youtu.be/7an5wU9Q5hk?t=393
 
https://youtu.be/7an5wU9Q5hk?t=393
  
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==Solution 1==
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==Video Solution 2==
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Let <math>k</math> be any positive integer so that <math>13k</math> is a multiple of <math>13</math>. For the smallest three-digit number, <math>13k>100</math> and <math>k>\frac{100}{13} \approx 7.7</math>. For the greatest three-digit number, <math>13k<999</math> and <math>k<\frac{999}{13} \approx 76.8</math>. The number <math>k</math> can range from <math>8</math> to <math>76</math> so there are <math>\boxed{\textbf{(C)}\ 69}</math> three-digit numbers.
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https://youtu.be/101Rgutx1R0 Soo, DRMS, NM
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==See Also==
 
==See Also==
 
{{AMC8 box|year=2005|num-b=17|num-a=19}}
 
{{AMC8 box|year=2005|num-b=17|num-a=19}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 03:07, 29 December 2022

Problem

How many three-digit numbers are divisible by 13?

$\textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77$


Solution

Let $k$ be any positive integer so that $13k$ is a multiple of $13$. For the smallest three-digit number, $13k>100$ and $k>\frac{100}{13} \approx 7.7$. For the greatest three-digit number, $13k<999$ and $k<\frac{999}{13} \approx 76.8$. The number $k$ can range from $8$ to $76$ so there are $\boxed{\textbf{(C)}\ 69}$ three-digit numbers.


Video Solution by OmegaLearn

https://youtu.be/7an5wU9Q5hk?t=393

Video Solution 2

https://youtu.be/101Rgutx1R0 Soo, DRMS, NM


See Also

2005 AMC 8 (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 AJHSME/AMC 8 Problems and Solutions

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