Difference between revisions of "2000 AMC 12 Problems/Problem 7"

Revision as of 01:04, 30 December 2020

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 $\log_{b} 729 = n$ , 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 $b \Longrightarrow \mathrm{E}$ .

2000 AMC 12 (Problems • Answer Key • 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
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 5: / Line 5: @@
 == Solution ==
-If <math>\displaystyle \log_{b} 729 = n</math>, then <math>b^n = 729</math>. Since <math>729 = 3^6</math>, <math>\displaystyle b</math> must be <math>3</math> to some [[factor]] of 6. Thus, there are four (3, 9, 27, 729) possible values of <math>\displaystyle b</math> <math>\Longrightarrow \mathrm{E}</math>.
+If <math>\log_{b} 729 = n</math>, then <math>b^n = 729</math>. Since <math>729 = 3^6</math>, <math>b</math> must be <math>3</math> to some [[factor]] of 6. Thus, there are four (3, 9, 27, 729) possible values of <math>b \Longrightarrow \mathrm{E}</math>.
 == See also ==
 {{AMC12 box|year=2000|num-b=6|num-a=8}}
 {{MAA Notice}}

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Difference between revisions of "2000 AMC 12 Problems/Problem 7"

Revision as of 01:04, 30 December 2020

Problem

Solution

See also