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# If x is an integer, is (x^2 + y^2)^(1/2) an integer?

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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
The required will be an integer if and only if (x^2 + y^2) is a perfect square. If (x^2 + y^2) is NOT a perfect square, then its square root will NOT be an integer.

Statement 1. x^2 = y^2 so we can say that x^2 + y^2 = 2x^2.

But 2x^2 Could be a perfect square if x = 0 (then square root of 2x^2 will be 0, which is an integer)

and for all other integer values of x, its NOT a perfect square (try putting any other integer value of x). so Insufficient.

Statement 2. x^2 - k^2 = -y^2
or x^2 + y^2 = k^2

Now k^2 will definitely be a perfect square since k is a positive integer, so its square root will be a positive integer.
Sufficient.

Hence B
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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
Bunuel wrote:
If x is an integer, is $$\sqrt{x^2 + y^2}$$ an integer?

(1) $$x^2 − y^2 = 0$$

(2) $$x^2 − k^2 = −y^2$$ for some positive integer k

Given $$\sqrt{x^2 + y^2}$$ ...we need to find this results in integer .

Stat 1: $$x^2 − y^2 = 0$$ => $$x^2 = y^2$$...Now substitute the $$y^2$$ value in equation then we get $$\sqrt{x^2 + x^2}$$..

From question stem we are given that x is an integer then we'll get $$\sqrt{2}$$*x...here x is an integer but $$\sqrt{2}$$ is not...since this is yes or no question ...we can say definte NO...Sufficient.

Stat 2: $$x^2 − k^2 = −y^2$$ => $$x^2 + y^2 = k^2$$...sub this value then we get $$\sqrt{k^2}$$....and since k is an integer the result in integer..Sufficient.

Option D.
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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
On the GMAT, can we have instances on a Yes/No DS question where one of the statements gives a No answer but the other gives a Yes ?
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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
Shruti0805 wrote:
On the GMAT, can we have instances on a Yes/No DS question where one of the statements gives a No answer but the other gives a Yes ?

No.

On the GMAT, two data sufficiency statements always provide TRUE information and these statements NEVER contradict each other or the stem. So, if you get a definite YES answer to the question from (1) you cannot get a definite NO answer from (2). Or say if you get that x = 2 only from (1) you cannot get x = 3 only from (2).
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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
Shruti0805 wrote:
On the GMAT, can we have instances on a Yes/No DS question where one of the statements gives a No answer but the other gives a Yes ?

Good question. As per GMAT history, I don't think such an instance has ever occurred. Experts here can guide us better.

And anyway, as I wrote in my explanation, answer should be B not D.
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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
amanvermagmat wrote:
Shruti0805 wrote:
On the GMAT, can we have instances on a Yes/No DS question where one of the statements gives a No answer but the other gives a Yes ?

Good question. As per GMAT history, I don't think such an instance has ever occurred. Experts here can guide us better.

And anyway, as I wrote in my explanation, answer should be B not D.

I answered this here: https://gmatclub.com/forum/if-x-is-an-i ... l#p1852776
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Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
stat1 : says x^2 = y^2
substitute in the stem,, suff,,,
given x is a integer,,

stat2 : clearly suff,,

ans D
Re: If x is an integer, is (x^2 + y^2)^(1/2) an integer? [#permalink]
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