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#### Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.  # If X & Y are Positive integers and X^2+Y^2 is odd, which of the..

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Senior Manager  P
Joined: 10 Jan 2017
Posts: 330
Location: India
If X & Y are Positive integers and X^2+Y^2 is odd, which of the..  [#permalink]

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Difficulty:   35% (medium)

Question Stats: 72% (01:14) correct 28% (01:21) wrong based on 18 sessions

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If X & Y are Positive integers and X^2+Y^2 is odd, which of the following is must be odd?

1>X+2y
2>3x+9Y
3>2X+Y
4>XY
5>XY-3X

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Intern  B
Joined: 15 Aug 2018
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GMAT 1: 710 Q49 V38 Re: If X & Y are Positive integers and X^2+Y^2 is odd, which of the..  [#permalink]

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J2S2019
If X & Y are Positive integers and X^2+Y^2 is odd, which of the following is must be odd?

We know that a positive no. is even then its square will also be even. Similarly, If a positive no. is odd then its square will also be odd.

We are given that X^2+Y^2 is odd.

This means that one of them is even and the other is odd. If both X and Y were Even/Odd, then the sum of their Squares would not be odd as odd+odd=even and even+even=even.

But keep in mind we don't know the signs X and Y. We can just deduce that when X is even, Y will be odd and When X is odd Y will be even.

Now we examine the options
1>X+2y
If X is even, then the expression becomes even

2>3x+9Y
Here both X and Y are multiplied with Odd numbers, So the even no. when multiplied with 3/9 will yield an even no. and the odd no. when multiplied with 3/9 will yield an odd no., Thus the overall expression will be Odd+Even= Odd.

3>2X+Y
If Y is even, then the expression will be even

4>XY
As we know one of them is even, So Even*odd=Even

5>XY-3X
We know XY is even from the previous case, further when X is even, XY -3X will be even.
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Re: If X & Y are Positive integers and X^2+Y^2 is odd, which of the..  [#permalink]

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There are a few tricks about even/odd nature of integers:
(1) EVEN * X --> always EVEN
(2) EVEN + X --> EVEN plays no role: if X is EVEN, the sum will always be EVEN; If X is ODD, the sum will be ODD.
(3) ODD * X --> ODD plays no role: if X is EVEN, the product will be EVEN; if X is ODD, the product will be ODD.

And the last, already mentioned by @adityapanda: power does not impact even-odd nature: X and X^2, X^3, X^4, etc. will have same even-odd nature. Let's ensure with example:
2 2^2=4 2^3=8... all powers of 2 are even
3 3^2=9 3^3=27... all powers of 3 are odd

Taking this into account, let's first analyze information from question stem:
X^2+Y^2 is odd, as power does not affect e-o nature, this implies that X+Y is odd, and this implies that X and Y are of different even-odd nature.

Taking (1)-(3) in mind, quickly looking on the answer choices: option B states 3x+9Y and we know that multiplication on ODD does not play any role, so 3X+9Y will have same even-odd nature as X+Y and we know from the information in question stem that X+Y is ODD. So, option B is correct answer choice.

Hope it helps. Re: If X & Y are Positive integers and X^2+Y^2 is odd, which of the..   [#permalink] 20 Oct 2019, 09:10
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