If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?

Question

If both x and y are integers, is  \sqrt{x^2 − y^2}  an integer?

(1) x + y = 0

(2) x^2 − y^2 equals the cube of an integer.

Source:   Nova GMAT  
Difficulty Level:   600

Kurtosis · Answer

St1:  x = -y  -->  \sqrt{(x^2 - y^2)} = 0 
Sufficient

St2:  x^2 - y^2 = 8 -->  \sqrt{(x^2 - y^2)}  is not an integer
 x^2 - y^2 = 64  -->  \sqrt{(x^2 - y^2)}  is an integer
Not Sufficient

Answer: A

Shruti0805 · Answer

\sqrt{x^2 − y^2}  =  \sqrt{(x+y)(x-y)} 
As per qsn., is  \sqrt{(x+y)(x-y)}  an integer ?

Stmt 1: x+y = 0
=>  \sqrt{x^2 − y^2}  =  \sqrt{0 * (x-y)}  = 0
Hence, sufficient.

Stmt. 2 :  x^2 − y^2 = I^3   
=>  \sqrt{x^2 − y^2}  =  \sqrt{I^3} 
Now, I can be 8 that would give the equation an integer value and can be 9 that would lead to a non integer value.
Not sufficient.

Answer should be A.

harsha3699 · Answer

We have both x and y are integers. 
We need to find out is (x^2-y^2)^1/2 is an integer or not. 
 Statement 1 says, x+y=0
From the question we can deduce as
(x^2-y^2)^1/2 = ((x+y)(x-y))^1/2
From statement 1 we get 
((x+y)(x-y))^1/2 = ((0)(x-y))^1/2
Implies, 0^1/2 = 0
So, yes (x^2-y^2)^1/2 = 0 (An integer) 
 Statement 1 is sufficient. 
 From Statement 2 we get that x^2-y^2 equals cube of an integer.
From this statement we can try x^2-y^2 with two of the most common cases 
 Case 1 , say x^2-y^2 = 27 which is 3^3 
So (x^2-y^2)^1/2 = (3^3)^1/2 = 3^3/2 (Not an integer) 
 Case 2 , say x^2-y^2 = 64 which is 4^3
So, (x^2-y^2)^1/2 = (4^3)^1/2 = 64^1/2 = 8 (An integer) 
 So we have two contradictory cases so, statement 2 is insufficient .
 So the correct option is A.

Mo2men · Answer

Let's analyze the expression:

\sqrt{x^2 − y^2}  =  \sqrt{(x+y)(x-y)}

(1) x + y = 0

The expression above will always equal to zero

Sufficient.

(2) x^2 − y^2 equals the cube of an integer

Let x^2 − y^2 =0 .......................Answer is yes

Let x^2 − y^2 =1 or -1 ...............Answer is yes

Let x^2 − y^2 =27 ......................Answer is No

Insufficient

P.S.:  I included different values to point put that we have quick easy values to prove yes/no answers.

Answer: A.

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If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?

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GMAT Data Sufficiency (DS) Questions

If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?

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