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

Question

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

(2) x is an odd integer.

niks18 · Accepted Answer

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

Nikhil · Answer

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

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

naturalimproviser · Answer

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

shivaniilall · Answer

Hello,

How come is it x^2 + x  + EVEN?

Shouldn't it be - EVEN?

Also, even - even = even. 
This means y^2 is even.
Which then should mean, y is even.

Please let me know if I have inferred this correctly.

yziyue · Answer

Statement 1 can be reduced to y(y+2) = x(x+1):
2y−x=x^2 - y^2
y^2 + 2y = x^2 + x
y(y+2) = x(x+1)

We can see x(x+1) will always be an even number, since one of the term here must be even. So y(y+2) must also be even. This allows us to confirm y is even since y and (y+2) will have the same even/odd properties.

unraveled · Answer

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

(1)  2y - x = x^2 - y^2 
Even - y = x + x^2

RHS will always be even thus, y is even.

SUFFICIENT.

(2) x is an odd integer.
Nothing about y or x's relation with y.

INSUFFICIENT.

Answer A.

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

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

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