Difference between revisions of "2009 AMC 10A Problems/Problem 9"
(New page: == Problem == Positive integers <math>a</math>, <math>b</math>, and <math>2009</math>, with <math>a<b<2009</math>, form a geometric sequence with an integer ratio. What is <math>a</math>?...) |
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== Solution == | == Solution == | ||
− | The prime factorization of <math>2009</math> is <math>2009 = 7\cdot 7\cdot 41</math>. As <math>a<b<2009</math>, the ratio must be positive and larger than <math>1</math>, hence there is only one possibility: the ratio must be <math>7</math>, and then <math>b=7\cdot 41</math>, and <math>a=\ | + | The prime factorization of <math>2009</math> is <math>2009 = 7\cdot 7\cdot 41</math>. As <math>a<b<2009</math>, the ratio must be positive and larger than <math>1</math>, hence there is only one possibility: the ratio must be <math>7</math>, and then <math>b=7\cdot 41</math>, and <math>a=41\Rightarrow\text{(B)}</math>. |
== See Also == | == See Also == | ||
{{AMC10 box|year=2009|ab=A|num-b=8|num-a=10}} | {{AMC10 box|year=2009|ab=A|num-b=8|num-a=10}} |
Revision as of 20:23, 2 November 2012
Problem
Positive integers , , and , with , form a geometric sequence with an integer ratio. What is ?
Solution
The prime factorization of is . As , the ratio must be positive and larger than , hence there is only one possibility: the ratio must be , and then , and .
See Also
2009 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
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 10 Problems and Solutions |