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# If y is a positive integer, is square_root of y

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If y is a positive integer, is square_root of y [#permalink]  30 Jun 2012, 02:46
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Question Stats:

57% (02:04) correct 43% (00:52) wrong based on 212 sessions
If y is a positive integer, is $$\sqrt{y}$$ an integer?

(1) $$\sqrt{4y}$$ is not an integer.
(2) $$\sqrt{5y}$$ is an integer.
[Reveal] Spoiler: OA

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Kudos [?]: 41109 [1] , given: 5666

Re: If y is a positive integer, is square_root of y [#permalink]  30 Jun 2012, 02:50
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metallicafan wrote:
If y is a positive integer, is $$\sqrt{y}$$ an integer?

(1) $$\sqrt{4y}$$ is not an integer.
(2) $$\sqrt{5y}$$ is an integer.

If $$y$$ is a positive integer is $$\sqrt{y}$$ an integer?

Note that as $$y$$ is a positive integer then $$\sqrt{y}$$ is either a positive integer or an irrational number. Also note that the question basically asks whether $$y$$ is a perfect square.

(1) $$\sqrt{4*y}$$ is not an integer --> $$\sqrt{4*y}=2*\sqrt{y}\neq{integer}$$ --> $$\sqrt{y}\neq{integer}$$. Sufficient.

(2) $$\sqrt{5*y}$$ is an integer --> $$y$$ can not be a prefect square because if it is, for example if $$y=x^2$$ for some positive integer $$x$$ then $$\sqrt{5*y}=\sqrt{5*x^2}=x\sqrt{5}\neq{integer}$$. Sufficient.

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Hope it helps.
_________________
Retired Moderator
Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
Joined: 04 Oct 2009
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Location: Peru
Schools: Harvard, Stanford, Wharton, MIT & HKS (Government)
WE 1: Economic research
WE 2: Banking
WE 3: Government: Foreign Trade and SMEs
Followers: 75

Kudos [?]: 439 [0], given: 109

Re: If y is a positive integer, is square_root of y [#permalink]  30 Jun 2012, 03:08
Bunuel wrote:
metallicafan wrote:
If y is a positive integer, is $$\sqrt{y}$$ an integer?

(1) $$\sqrt{4y}$$ is not an integer.
(2) $$\sqrt{5y}$$ is an integer.

If $$y$$ is a positive integer is $$\sqrt{y}$$ an integer?

Note that as $$y$$ is a positive integer then $$\sqrt{y}$$ is either a positive integer or an irrational number. Also note that the question basically asks whether $$y$$ is a perfect square.

(1) $$\sqrt{4*y}$$ is not an integer --> $$\sqrt{4*y}=2*\sqrt{y}\neq{integer}$$ --> $$\sqrt{y}\neq{integer}$$. Sufficient.

(2) $$\sqrt{5*y}$$ is an integer --> $$y$$ can not be a prefect square because if it is, for example if $$y=x^2$$ for some positive integer $$x$$ then $$\sqrt{5*y}=\sqrt{5*x^2}=x\sqrt{5}\neq{integer}$$. Sufficient.

Hope it helps.

Thank you Bunuel. I arrived to the same conclusion. However, I have a doubt:
Ok, we know that $$x\sqrt{5}$$.
But how can we be so sure that x is not for example a a huge number like 10*10^10000000.... to make $$x\sqrt{5}$$ a integer?
_________________

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Kudos [?]: 41109 [0], given: 5666

Re: If y is a positive integer, is square_root of y [#permalink]  30 Jun 2012, 03:16
Expert's post
metallicafan wrote:
Bunuel wrote:
metallicafan wrote:
If y is a positive integer, is $$\sqrt{y}$$ an integer?

(1) $$\sqrt{4y}$$ is not an integer.
(2) $$\sqrt{5y}$$ is an integer.

If $$y$$ is a positive integer is $$\sqrt{y}$$ an integer?

Note that as $$y$$ is a positive integer then $$\sqrt{y}$$ is either a positive integer or an irrational number. Also note that the question basically asks whether $$y$$ is a perfect square.

(1) $$\sqrt{4*y}$$ is not an integer --> $$\sqrt{4*y}=2*\sqrt{y}\neq{integer}$$ --> $$\sqrt{y}\neq{integer}$$. Sufficient.

(2) $$\sqrt{5*y}$$ is an integer --> $$y$$ can not be a prefect square because if it is, for example if $$y=x^2$$ for some positive integer $$x$$ then $$\sqrt{5*y}=\sqrt{5*x^2}=x\sqrt{5}\neq{integer}$$. Sufficient.

Hope it helps.

Thank you Bunuel. I arrived to the same conclusion. However, I have a doubt:
Ok, we know that $$x\sqrt{5}$$.
But how can we be so sure that x is not for example a a huge number like 10*10^10000000.... to make $$x\sqrt{5}$$ a integer?

The point is that $$\sqrt{5}$$ is an irrational number, and the decimal representation of an irrational number never repeats or terminates (irrational numbers are not terminating decimals). So, $$integer*irrational\neq{integer}$$ (no matter how large x is, $$x\sqrt{5}$$ will never be an integer).

Hope it's clear.
_________________
Retired Moderator
Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
Joined: 04 Oct 2009
Posts: 1725
Location: Peru
Schools: Harvard, Stanford, Wharton, MIT & HKS (Government)
WE 1: Economic research
WE 2: Banking
WE 3: Government: Foreign Trade and SMEs
Followers: 75

Kudos [?]: 439 [0], given: 109

Re: If y is a positive integer, is square_root of y [#permalink]  30 Jun 2012, 17:22
Bunuel wrote:
The point is that $$\sqrt{5}$$ is an irrational number, and the decimal representation of an irrational number never repeats or terminates (irrational numbers are not terminating decimals). So, $$integer*irrational\neq{integer}$$ (no matter how large x is, $$x\sqrt{5}$$ will never be an integer).

Hope it's clear.

Thank you Bunuel! An additional question: all the square roots (or roots in general) of prime numbers are irrational?
Thanks!
_________________

"Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can."

My Integrated Reasoning Logbook / Diary: my-ir-logbook-diary-133264.html

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Math Expert
Joined: 02 Sep 2009
Posts: 27228
Followers: 4231

Kudos [?]: 41109 [0], given: 5666

Re: If y is a positive integer, is square_root of y [#permalink]  01 Jul 2012, 01:24
Expert's post
metallicafan wrote:
Bunuel wrote:
The point is that $$\sqrt{5}$$ is an irrational number, and the decimal representation of an irrational number never repeats or terminates (irrational numbers are not terminating decimals). So, $$integer*irrational\neq{integer}$$ (no matter how large x is, $$x\sqrt{5}$$ will never be an integer).

Hope it's clear.

Thank you Bunuel! An additional question: all the square roots (or roots in general) of prime numbers are irrational?
Thanks!

Since no prime can be a perfect square (or perfect cube, ...), then $$\sqrt{prime}=irrational$$.
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Kudos [?]: 41109 [1] , given: 5666

Re: If y is a positive integer, is square_root of y [#permalink]  12 Jun 2013, 03:27
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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: If y is a positive integer, is square_root of y   [#permalink] 12 Jun 2013, 03:27
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