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

Question

1)  8y^2+x^2=x^4+y^4 
2)  y=6-x

A modified version of a Q on GMATCLUB chat..

AMahmoodi · Answer

Hi,

st1. ---> we can cross x from both sides and ignore all of the powers as powers don't change the even or odd status . we get 8y==y,so the left side is an even int because 8 is even therefore the right side must be even. insuff.

st2. y=6-x, x could be either even or odd depending on y. insuff.

1+2) sufficient. y is even from 1 therefore x must be even too(st2).

CholericQuill · Answer

Statement 1 implies y must be even but is insufficient to conclude anything about x.
Explanation -  8y^2  is even. 
Case 1 - x is even, therefore  x^2  and  x^4  are even. LHS is even, for RHS to be even,  y^4  must be even, thus y is even.
Case 2 - x is odd, then  x^2  and  x^4  are odd. LHS is odd, and  y^4  must be even for RHS to be odd.

Not sufficient

Statement -2 
x + y = 6 implies either both are even or both are odd. Not sufficient.

Statement 1 + 2
y is even from (1), thus x is even from (2)

Answer - C

bumpbot · Answer

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

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