Difference between revisions of "2003 AMC 10A Problems/Problem 8"

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== Problem ==
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#REDIRECT[[2003 AMC 12A Problems/Problem 8]]
What is the probability that a randomly drawn positive factor of <math>60</math> is less than <math>7</math>
 
 
 
<math> \mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{6}\qquad \mathrm{(C) \ } \frac{1}{4}\qquad \mathrm{(D) \ } \frac{1}{3}\qquad \mathrm{(E) \ } \frac{1}{2} </math>
 
 
 
== Solution ==
 
For a positive number <math>n</math> which is not a perfect square, exactly half of the positive factors will be less than <math>\sqrt{n}</math>.
 
 
 
Since <math>60</math> is not a perfect square, half of the positive factors of <math>60</math> will be less than <math>\sqrt{60}\approx 7.746</math>.
 
 
 
Clearly, there are no positive factors of <math>60</math> between <math>7</math> and <math>\sqrt{60}</math>.
 
 
 
Therefore half of the positive factors will be less than <math>7</math>.
 
 
 
So the answer is <math>\frac{1}{2} \Rightarrow E</math>.
 
 
 
== See Also ==
 
{{AMC10 box|year=2003|ab=A|num-b=7|num-a=9}}
 
 
 
[[Category:Introductory Number Theory Problems]]
 

Latest revision as of 16:54, 31 July 2011

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