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

(Created page with "==Problem== How many three-digit numbers are divisible by 13? <math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 67\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 76\qquad\textbf{(E)}\ 77</mat...")
 
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==See Also==
 
{{AMC8 box|year=2005|num-b=17|num-a=19}}
 
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Revision as of 00:10, 5 July 2013

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.

See Also

2005 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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All AJHSME/AMC 8 Problems and Solutions

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