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2000 AMC 12 Problems/Problem 7

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Problem

How many positive integers $\displaystyle b$ have the property that $\displaystyle \log_{b} 729$ is a positive integer?

$\mathrm{(A) \ 0 } \qquad \mathrm{(B) \ 1 } \qquad \mathrm{(C) \ 2 } \qquad \mathrm{(D) \ 3 } \qquad \mathrm{(E) \ 4 }$

Solution

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

See also

2000 AMC 12 (Problems • Resources)
Preceded by Problem 6	Followed by Problem 8
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

Retrieved from "http://www.artofproblemsolving.com/Wiki/index.php/2000_AMC_12_Problems/Problem_7"

Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

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