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Difference between revisions of "2002 AMC 12P Problems/Problem 4"

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== Problem ==
 
== Problem ==
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How many positive [[integer]]s <math>b</math> have the property that <math>\log_{b} 729</math> is a positive integer?
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Let <math>a</math> and <math>b</math> be distinct real numbers for which
 +
<cmath>\frac{a}{b} + \frac{a+10b}{b+10a} = 2.</cmath>
  
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<math> \mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 } </math>
+
Find <math>\frac{a}{b}</math>
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<math>
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\text{(A) }0.4
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\qquad
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\text{(B) }0.5
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\qquad
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\text{(C) }0.6
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\qquad
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\text{(D) }0.7
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\qquad
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\text{(E) }0.8
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</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 00:40, 30 December 2023

Problem

Let $a$ and $b$ be distinct real numbers for which \[\frac{a}{b} + \frac{a+10b}{b+10a} = 2.\]

Find $\frac{a}{b}$

$\text{(A) }0.4 \qquad \text{(B) }0.5 \qquad \text{(C) }0.6 \qquad \text{(D) }0.7 \qquad \text{(E) }0.8$

Solution

If $\log_{b} 729 = n$, then $b^n = 729$. Since $729 = 3^6$, $b$ must be $3$ to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of $b \Longrightarrow \boxed{\mathrm{E}}$.

See also

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by
Problem 3 		Followed by
Problem 5
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 AMC 12 Problems and Solutions

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