Difference between revisions of "2005 AIME I Problems/Problem 3"

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== Solution ==
 
== Solution ==
<math>n=p\cdot q</math> or <math>n=p^3</math> where p and q are distinct primes. In the first case, the three proper divisors of n are 1, p, and q. Because there are 15 prime numbers less than 50, there are <math> {15 \choose 2} =105</math> numbers of the first type.There are four integers of the second type because 2, 3, 5, and 7 are the only primes with squares less than 50. Thus there are <math>105+4=109</math> integers that meet the given conditions.
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Suppose <math>n</math> is such an [[integer]].  Then <math>n=p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math>.  
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In the first case, the three proper divisors of <math>n</math> are 1, <math>p</math> and <math>q</math>. Thus, we need to pick two prime numbers less than 50.  There are fifteen of these (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47) so there are <math> {15 \choose 2} =105</math> numbers of the first type.
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In the second case, the three proper divisors of <math>n</math> are 1, <math>p</math> and <math>p^2</math>.  Thus we need to pick a prime number whose square is less than 50.  There are four of these (2, 3, 5 and 7) and so four numbers of the second type.  
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Thus there are <math>105+4=109</math> integers that meet the given conditions.
  
 
== See also ==
 
== See also ==
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* [[2005 AIME I Problems/Problem 4 | Next problem]]
 
* [[2005 AIME I Problems/Problem 4 | Next problem]]
 
* [[2005 AIME I Problems]]
 
* [[2005 AIME I Problems]]
 
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* [[Divisor_function#Demonstration | Counting divisors of positive integers]]
 
[[Category:Introductory Number Theory Problems]]
 
[[Category:Introductory Number Theory Problems]]

Revision as of 12:43, 29 November 2006

Problem

How many positive integers have exactly three proper divisors, each of which is less than 50?

Solution

Suppose $n$ is such an integer. Then $n=p\cdot q$ or $n=p^3$ for distinct prime numbers $p$ and $q$.

In the first case, the three proper divisors of $n$ are 1, $p$ and $q$. Thus, we need to pick two prime numbers less than 50. There are fifteen of these (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47) so there are ${15 \choose 2} =105$ numbers of the first type.

In the second case, the three proper divisors of $n$ are 1, $p$ and $p^2$. Thus we need to pick a prime number whose square is less than 50. There are four of these (2, 3, 5 and 7) and so four numbers of the second type.

Thus there are $105+4=109$ integers that meet the given conditions.

See also

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