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# If x is a positive integer, is x an integer? (1) 4x is an

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If x is a positive integer, is x an integer? (1) 4x is an [#permalink]  29 Sep 2010, 07:09
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If x is a positive integer, is \sqrt{x} an integer?
(1) \sqrt{4x} is an integer.
(2) \sqrt{3x} is not an integer.

I am not satisfied with the official explanation. Please give yours, thanks.

Edited, thanks
[Reveal] Spoiler: OA

Last edited by ezinis on 29 Sep 2010, 09:39, edited 1 time in total.
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Re: Q 31, OG 12 DS [#permalink]  29 Sep 2010, 07:29
ezinis wrote:
If x is a positive integer, is \sqrt{x} an integer?
(1) \sqrt{4x} is an integer4.
(2) \sqrt{3x} is an integer.

I am not satisfied with the official explanation. Please give yours, thanks.

I think (2) should be \sqrt{3x} is NOT an integer.

If x=integer, is \sqrt{x}=integer?

(1) \sqrt{4x} is an integer --> 2\sqrt{x}=integer --> 2\sqrt{x} to be an integer \sqrt{x} must be an integer or integer/2, but as x is an integer, then \sqrt{x} can not be integer/2, hence \sqrt{x} is an integer. Sufficient.

(2)\sqrt{3x} is not an integer --> if x=9, condition \sqrt{3x}=\sqrt{27} is not an integer satisfied and \sqrt{x}=3 IS an integer, BUT if x=2, condition \sqrt{3x}=\sqrt{6} is not an integer satisfied and \sqrt{x}=\sqrt{2} IS NOT an integer. Two different answers. Not sufficient.

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Re: Q 31, OG 12 DS [#permalink]  29 Sep 2010, 08:10
ezinis wrote:
If x is a positive integer, is \sqrt{x} an integer?
(1) \sqrt{4x} is an integer4.
(2) \sqrt{3x} is an integer.

I am not satisfied with the official explanation. Please give yours, thanks.

(1) \sqrt{4x} = 2 * \sqrt{x}
If this is an integer, then \sqrt{x} has to be an integer

(2) \sqrt{3x} = \sqrt{3} * \sqrt{x}
For this to be an integer, \sqrt{x} must be of the form \sqrt{3} * Integer
So \sqrt{x} is not an integer

I am not sure if the question is correct as (1) and (2) are contradicting. Is it supposed to say \sqrt{3x} is not an integer ?
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Re: Q 31, OG 12 DS   [#permalink] 29 Sep 2010, 08:10
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