Maksym wrote:

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

(1) 4y^2+3x^2=x^4+y^4

(2) y=4−x^2

The official answer is A, and the logic is clear to me.

But, is it possible in the first equations also to have x = y = o? Shouldn't the answer be E in such case?

It is not explicitly stated in the wording that x and y are different non-zero integers?

Most probably I just missed smth, so would be grateful for your explanations

Thanks in advance!

M27-02

If x and y are integers, is y an even integer?(1) 4y^2+3x^2=x^4+y^4 --> rearrange: \(3x^2-x^4=y^4-4y^2\) --> \(x^2(3-x^2)=y^2(y^2-4)\). Notice that LHS is even for any value of \(x\): if \(x\) is odd then \(3-x^2=odd-odd=even\) and if \(x\) is even then the product is naturally even. So, \(y^2(y^2-4)\) is also even, but in order it to be even \(y\)

must be even, since if \(y\) is odd then \(y^2(y^2-4)=odd*(odd-even)=odd*odd=odd\). Sufficient.

(2) y=4-x^2 --> if \(x=odd\) then \(y=even-odd=odd\) but if \(x=even\) then \(y=even-even=even\). Not sufficient.

Answer: A.As for your doubt: 0 is an even integer. An even number is an

integer that is "evenly divisible" by 2, i.e., divisible by 2 without a remainder and as zero is evenly divisible by 2 then it must be even.

Check more tips on zero and number properties here:

tips-and-hints-for-specific-quant-topics-with-examples-172096.html#p1371030Hope it helps.