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Math Expert V
Joined: 02 Sep 2009
Posts: 55266
If x and y are integers, is y an even integer?  [#permalink]

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Difficulty:   75% (hard)

Question Stats: 47% (01:52) correct 53% (01:49) wrong based on 105 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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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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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?  [#permalink]

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It is a Yes/No question, asking if Y is an even integer.

Let's start with statement 2, it says X is an odd integer, no information is provided regarding Y. Hence Insufficient.
You can eliminate options B and D.

Statement 1: -
2y−x=x^2 - y^2
y^2 + 2y = x^2 + x
y(y+2) = x(x+1)

Now we have four possibilities,
x,y - Both Even
x - Even, y - Odd
x - Odd, Y - Even
x,y - Both Odd

The equation works only in the first and the third case, that is when either both x and y are even or when x is odd and y is even. We are only concerned about y being even. Hence, Sufficient - A
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Re: If x and y are integers, is y an even integer?  [#permalink]

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_________________ Re: If x and y are integers, is y an even integer?   [#permalink] 10 Apr 2019, 04:07
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