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# If n is a positive integer and is not the square of any

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Joined: 01 Apr 2008
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Name: Ronak Amin
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If n is a positive integer and is not the square of any [#permalink]

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05 Oct 2009, 09:45
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If n is a positive integer and is not the square of any integer, is it a prime number?
1). Among the factors of n, only n is greater than n^0.5
2). Among the factors of n, only 1 is less than n^0.5
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Joined: 11 Sep 2009
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05 Oct 2009, 11:15

We are told that n is NOT the square of any integer. We need to determine if n is a prime number.

Statement 1: Among the factors of n, only n is greater than n^0.5.

All integers have "factor pairs", which when multiplied together, result in the product of the original number. For example, 12 has factor pairs of 12 and 1, 6 and 2, and 4 and 3. These are always two distinct numbers unless the number is a square (25 = 5 * 5). These factor pairs, however, will always have one number which is greater than the square root of the number, and one number which is less than the square root of the number.

Since n does not have any factors greater than n^0.5 (other than itself) it is a prime number. SUFFICIENT.

Statement 2: Among the factors of n, only 1 is less than n^0.5 .

Since n does not have any factors less than n^0.5 (other than 1) it is a prime number. SUFFICIENT.

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05 Oct 2009, 11:25
If integer n>1 has no factors less then or equal to n^(1/2) n is prime.

(1) Among the factors of n, only n is greater than n^0.5 --> no factor less then n^(1/2) --> n prime
(2) Among the factors of n, only 1 is less than n^0.5 --> no factor less then n^(1/2) --> n prime

D.
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Re: Prime?   [#permalink] 05 Oct 2009, 11:25
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