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If x and y are integers and x 2 + y 2 is odd, which of the following m

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Carcass
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If x and y are integers and x 2 + y 2 is odd, which of the following m [#permalink] New post   15 Sep 2021, 01:16
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If x and y are integers and \(x^2 + y^2\) is odd, which of the following must be even?

(A) x
(B) y
(C) x + y
(D) xy + y
(E) xy
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E
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Re: If x and y are integers and x 2 + y 2 is odd, which of the following m [#permalink] New post   15 Sep 2021, 09:35
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\(x^2 + y^2\) is odd.

So we can say for sure that one of them will be even and the other will be odd. No other possibilities.

Hence, the multiplication of both will be even.

Answer E
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Re: If x and y are integers and x 2 + y 2 is odd, which of the following m [#permalink] New post   15 Sep 2021, 11:14
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Saying \(x^2+y^2\) is odd is same as \(x+y\) is odd. Since squaring keeps the parity.
if \(x+y\) is odd, then only one of them is odd, which means \(odd * even\) = even, so E)
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Re: If x and y are integers and x 2 + y 2 is odd, which of the following m [#permalink] New post   15 Sep 2021, 18:51
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The problem can be simplified as x+y is odd, since odd*odd=odd and even*even=even. It's not known whether x or y is even or odd, but one must be even and the other is odd. Hence, only option E is correct and matches our condition set.

Option D is not appropriate, as odd*even can be added by both even or odd, resulting in even and odd accordingly.

Carcass wrote:
If x and y are integers and \(x^2 + y^2\) is odd, which of the following must be even?

(A) x
(B) y
(C) x + y
(D) xy + y
(E) xy
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