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

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Math Expert
Joined: 02 Sep 2009
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If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?  [#permalink]

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15 May 2017, 10:26
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65% (01:15) correct 35% (01:16) wrong based on 74 sessions

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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.

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Re: If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?  [#permalink]

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15 May 2017, 20:21
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

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Re: If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?  [#permalink]

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15 May 2017, 20:34
Bunuel wrote:
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.

$$\sqrt{x^2 − y^2}$$ = $$\sqrt{(x+y)(x-y)}$$
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 : $$x^2 − y^2 = I^3$$ [I= Integer]
=> $$\sqrt{x^2 − y^2}$$ = $$\sqrt{I^3}$$
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.

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Joined: 21 Nov 2016
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Re: If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?  [#permalink]

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30 Sep 2018, 20:07
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.
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Joined: 26 Mar 2013
Posts: 2068
Re: If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?  [#permalink]

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01 Oct 2018, 03:44
Bunuel wrote:
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.

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.

Re: If both x and y are integers, is (x^2 − y^2)^(1/2) an integer?   [#permalink] 01 Oct 2018, 03:44
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