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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https://youtu.be/7an5wU9Q5hk?t=393
  
 
==Solution 1==
 
==Solution 1==
 
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.
 
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.
 
==Solution 2==
 
EVERYONE LOVES <math>69</math> and thus that's the answer
 
  
 
==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 19:02, 27 October 2020

Problem

How many three-digit numbers are divisible by 13?

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

Video Solution

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

Solution 1

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.

See Also

2005 AMC 8 (ProblemsAnswer KeyResources)
Preceded by
Problem 17
Followed by
Problem 19
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All AJHSME/AMC 8 Problems and Solutions

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