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# If x and y are integers, is y an even integer?

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Math Expert
Joined: 02 Sep 2009
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If x and y are integers, is y an even integer?  [#permalink]

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10 Jul 2018, 20:59
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Difficulty:

65% (hard)

Question Stats:

47% (01:43) correct 53% (01:31) wrong based on 30 sessions

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If x and y are integers, is y an even integer?

(1) $$2y - x = x^2 - y^2$$

(2) x is an odd integer.

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If x and y are integers, is y an even integer?  [#permalink]

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10 Jul 2018, 21:59
1
Bunuel wrote:
If x and y are integers, is y an even integer?

(1) $$2y - x = x^2 - y^2$$

(2) x is an odd integer.

Statement 1: $$2y - x = x^2 - y^2$$. $$2y$$ will always be Even

$$=>y^2=x^2+x+Even$$. Now irrespective of the value of $$x$$, $$x^2+x$$ will always be Even, because $$x$$ is an integer.

Hence $$y^2=Even+Even=Even$$. Sufficient

Statement 2: nothing mentioned about $$y$$. Insufficient

Option A
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Re: If x and y are integers, is y an even integer?  [#permalink]

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10 Jul 2018, 22:05
GIven X and Y are integers

We need to find if Y is an even integer

Statement 1

$$2y - x = x^{2}-y^{2}$$

=> $$2y = x^{2}-y^{2} + x$$

The left hand side of above equation is always even

=> $$x^{2}-y^{2} + x$$ is even

Lets see the result for different combinations of X and Y

X | Y | $$x^{2}-y^{2} + x$$

Even | Even | Even
Odd | Even | Even
Odd | Odd | Odd
Even | Odd | Odd

We can see that Y is an even integer whenever $$x^{2}-y^{2} + x$$ is even

Statement 1 is sufficient

Statement 2

X is an odd integer => doesn't tell anything about Y

Statement 2 not sufficient

Analysis of statement 1 and 2 together is not required

Hence option A
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Re: If x and y are integers, is y an even integer? &nbs [#permalink] 10 Jul 2018, 22:05
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