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

Question

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

gmatexam439 · Answer

D

1. root 2*x will never be an integer
sufficient

2.root K^2 will always be an integer
sufficient

sauravleo123 · Answer

x^2-y^2=0
so x=y or x=-y
so we get root 2 * y
We dont know y
if y = root 2
then above expression = int
if not then no
Insufficient
Statement 2 is obviously sufficient
B

amanvermagmat · Answer

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

msk0657 · Answer

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.

Shruti0805 · Answer

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 ?

Bunuel · Answer

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).

amanvermagmat · Answer

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.

Bunuel · Answer

I answered this here: https://gmatclub.com/forum/if-x-is-an-i ... l#p1852776

mohshu · Answer

stat1 : says x^2 = y^2
 substitute in the stem,, suff,,,
 given x is a integer,,

stat2 : clearly suff,,

ans D

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

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