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

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== Problem == | == Problem == | ||

− | + | What is the probability that a randomly drawn positive factor of <math>60</math> is less than <math>7</math> | |

− | <math> \mathrm{(A) \ } | + | <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 == | == 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 | + | Therefore half of the positive factors will be less than <math>7</math>. |

− | + | So the answer is <math>\frac{1}{2} \Rightarrow E</math>. | |

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== See Also == | == See Also == | ||

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*[[2003 AMC 10A Problems/Problem 9|Next Problem]] | *[[2003 AMC 10A Problems/Problem 9|Next Problem]] | ||

− | [[Category:Introductory | + | [[Category:Introductory Number Theory Problems]] |

## Revision as of 20:08, 4 November 2006

## Problem

What is the probability that a randomly drawn positive factor of is less than

## Solution

For a positive number which is not a perfect square, exactly half of the positive factors will be less than .

Since is not a perfect square, half of the positive factors of will be less than .

Clearly, there are no positive factors of between and .

Therefore half of the positive factors will be less than .

So the answer is .