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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]

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New post 30 Jun 2012, 02:46
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If y is a positive integer, is \(\sqrt{y}\) an integer?

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

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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 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.

Answer: D.

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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 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.

Answer: D.

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?
Maybe, I am forgeting a concept. Please, your help.
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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 30 Jun 2012, 03:16
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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.

Answer: D.

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?
Maybe, I am forgeting a concept. Please, your help.


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.
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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 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!
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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 01 Jul 2012, 01:24
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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Re: If y is a positive integer, is square_root of y  [#permalink]

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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 13 Aug 2015, 14:04
Bunuel

What if y=125?

125 is an integer.

sqrt of 5*125 = sqrt of 625 = 25

This would make B insufficient.

Am I missing something here?
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Re: If y is a positive integer, is square_root of y  [#permalink]

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New post 14 Aug 2015, 16:11
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Mascarfi wrote:
Bunuel

What if y=125?

125 is an integer.

sqrt of 5*125 = sqrt of 625 = 25

This would make B insufficient.

Am I missing something here?


No, your calculations are fine and are consistent with the question. If you have y = 125, \(\sqrt{125}\) \(\neq\) Integer and thus get an unambiguous "no" making statement 2 sufficient. The original question is

"\(\sqrt{y}\) an integer?" So getting an unambiguous yes or no will be sufficient. Per Bunuel's solution, it is shown that y can not be a perfect square leading to \(\sqrt{y}\neq Integer\)

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If y is a positive integer, is square_root of y  [#permalink]

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New post 14 Aug 2015, 16:48
It's clear now. Completely forgot what the question was asking while looking at number 2. thanks
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Re: If y is a positive integer, is square_root of y  [#permalink]

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