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

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Director
Joined: 01 Apr 2008
Posts: 875

Kudos [?]: 843 [0], given: 18

Name: Ronak Amin
Schools: IIM Lucknow (IPMX) - Class of 2014
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

Kudos [?]: 843 [0], given: 18

Manager
Joined: 11 Sep 2009
Posts: 129

Kudos [?]: 423 [0], given: 6

Re: Prime? [#permalink]

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05 Oct 2009, 11:15
The answer is D.

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.

Therefore the answer is D.

Kudos [?]: 423 [0], given: 6

Math Expert
Joined: 02 Sep 2009
Posts: 41891

Kudos [?]: 128901 [0], given: 12183

Re: Prime? [#permalink]

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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.
_________________

Kudos [?]: 128901 [0], given: 12183

Re: Prime?   [#permalink] 05 Oct 2009, 11:25
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# If n is a positive integer and is not the square of any

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