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# If x is a positive integer, is x an integer?

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If x is a positive integer, is x an integer? [#permalink]

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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.
[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]

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29 Sep 2010, 07:29
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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]

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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: If x is a positive integer, is x an integer? [#permalink]

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09 Feb 2014, 09:55
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Re: If x is a positive integer, is x an integer? [#permalink]

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27 Sep 2015, 08:36
Bunuel wrote:
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.

i think it does not matter whether statement 2 is integer or not, as in both the cases we are getting different solution. and answer will be one in both the cases.
Re: If x is a positive integer, is x an integer?   [#permalink] 27 Sep 2015, 08:36
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