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Difference between revisions of "1997 AHSME Problems/Problem 21"

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==Problem==
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For any positive integer <math>n</math>, let
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<math> f(n) =\left\{\begin{matrix}\log_{8}{n}, &\text{if }\log_{8}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right. </math>
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What is <math> \sum_{n = 1}^{1997}{f(n)} </math>?
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<math> \textbf{(A)}\ \log_{8}{2047}\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ \frac{55}{3}\qquad\textbf{(D)}\ \frac{58}{3}\qquad\textbf{(E)}\ 585 </math>
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==Solution==
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For positive integers <math>n</math>, <math>\log_8 n</math> is rational if and only if <math>n = 2^k</math> for an integer <math>k</math>.  That's because <math>\log_8 2^k = k\log_8 2 = \frac{k}{3}</math>.
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So we actually want to find <math>\sum_{k=0}^{10} \log_8 2^k</math>, since <math>2^{11}</math> will be over <math>1997</math>.
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Using log properties, we get <math>\sum_{k=0}^{10} k \log_8 2</math>
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<math>\frac{1}{3}\sum_{k=0}^{10} k</math>
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<math>\frac{1}{3}\cdot (\frac{10\cdot 11}{2})</math>
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<math>\frac{55}{3}</math>, and the answer is <math>\boxed{C}</math>
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== See also ==
 
== See also ==
 
{{AHSME box|year=1997|num-b=20|num-a=22}}
 
{{AHSME box|year=1997|num-b=20|num-a=22}}
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{{MAA Notice}}

Latest revision as of 13:13, 5 July 2013

Problem

For any positive integer $n$, let

$f(n) =\left\{\begin{matrix}\log_{8}{n}, &\text{if }\log_{8}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.$

What is $\sum_{n = 1}^{1997}{f(n)}$?

$\textbf{(A)}\ \log_{8}{2047}\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ \frac{55}{3}\qquad\textbf{(D)}\ \frac{58}{3}\qquad\textbf{(E)}\ 585$

Solution

For positive integers $n$, $\log_8 n$ is rational if and only if $n = 2^k$ for an integer $k$. That's because $\log_8 2^k = k\log_8 2 = \frac{k}{3}$.

So we actually want to find $\sum_{k=0}^{10} \log_8 2^k$, since $2^{11}$ will be over $1997$.

Using log properties, we get $\sum_{k=0}^{10} k \log_8 2$

$\frac{1}{3}\sum_{k=0}^{10} k$

$\frac{1}{3}\cdot (\frac{10\cdot 11}{2})$

$\frac{55}{3}$, and the answer is $\boxed{C}$

See also

1997 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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