Difference between revisions of "2000 AMC 12 Problems/Problem 7"
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== Solution == | == 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>\ | + | 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>. |
== See also == | == See also == | ||
Revision as of 17:18, 4 March 2007
Problem
How many positive integers 
have the property that 
is a positive integer?
Solution
If 
, then 
. Since 
, must be 
to some factor of 6. Thus, there are four (3, 9, 27, 729) possible values of 
.
See also
| 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 | |
