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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
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Schools: IIM Lucknow (IPMX) - Class of 2014
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If n is a positive integer and is not the square of any [#permalink]  05 Oct 2009, 08:45
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0% (00:00) correct 100% (01:15) wrong based on 2 sessions
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
Manager
Joined: 11 Sep 2009
Posts: 129
Followers: 4

Kudos [?]: 216 [0], given: 6

Re: Prime? [#permalink]  05 Oct 2009, 10:15

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.

Math Expert
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Re: Prime? [#permalink]  05 Oct 2009, 10:25
Expert's post
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.
_________________
Re: Prime?   [#permalink] 05 Oct 2009, 10:25
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