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# If positive integers x and y are not both odd, which of the following

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Intern
Joined: 26 Feb 2011
Posts: 29
If positive integers x and y are not both odd, which of the following  [#permalink]

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Updated on: 15 May 2016, 04:01
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If positive integers x and y are not both odd, which of the following must be even?

(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-positive-integers-x-and-y-are-not-both-odd-which-of-the-165827.html

Originally posted by Madelaine88 on 01 Mar 2011, 08:20.
Last edited by Bunuel on 15 May 2016, 04:01, edited 1 time in total.
Renamed the topic and edited the question.
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Posts: 1504
Re: If positive integers x and y are not both odd, which of the following  [#permalink]

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01 Mar 2011, 08:42
means either one of x and y is even or both x and y are even.

First case:
where one is even;
xy = even; even multiplied by odd is even.
where both are even;
xy = even; even multiplied by even is even.

so; "A" must be even.

$$\frac{x+y}{x-y}$$

These rules don't apply to division.

Say one of x and y is even.
x+y = odd
x-y = odd

odd/odd can be odd or non-integer but never even.

Say both x and y are even
x+y = even
x-y = even

even/even can be odd, even or non-integer.
(12+4)/(12-4) = 2 even
(8+4)/(8-4) = 3 odd
(12+2)/(12-2) = 1.4 non integer
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Posts: 2015
Re: If positive integers x and y are not both odd, which of the following  [#permalink]

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01 Mar 2011, 10:54
1
If positive integers x and y are not both odd, which of the following must be even?
a/ xy
b/ x+y
c/ x-y
d/ x+y-1
e/ 2(x+y)-1

Why x+y/x-y cant be even?

fluke, I think Madelaine was asking why x+y or x-y is not the correct answer here (I don't think the question was about division, since (x+y)/(x-y) is not among the answer choices). It's vital here to note that the question asks "which of the following *must* be even". That's a very different question from "which of the following *could* be even". Certainly x+y *could* be even, if x and y are both even, but if, say, x is even and y is odd, then x+y will be odd. So x+y does not need to be even, and B is thus not the right answer here. But if at least one of x or y is even, then xy will always be even, so the answer is A.
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Re: If positive integers x and y are not both odd, which of the following  [#permalink]

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01 Mar 2011, 11:07
IanStewart wrote:
If positive integers x and y are not both odd, which of the following must be even?
a/ xy
b/ x+y
c/ x-y
d/ x+y-1
e/ 2(x+y)-1

Why x+y/x-y cant be even?

fluke, I think Madelaine was asking why x+y or x-y is not the correct answer here (I don't think the question was about division, since (x+y)/(x-y) is not among the answer choices). It's vital here to note that the question asks "which of the following *must* be even". That's a very different question from "which of the following *could* be even". Certainly x+y *could* be even, if x and y are both even, but if, say, x is even and y is odd, then x+y will be odd. So x+y does not need to be even, and B is thus not the right answer here. But if at least one of x or y is even, then xy will always be even, so the answer is A.

I see!!! Now, that makes sense! I wondered that for a second and then carried on. thanks IanStewart.
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Re: If positive integers x and y are not both odd, which of the following  [#permalink]

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02 Mar 2011, 01:02
thanks IanStewart. Nicely explained.
Intern
Joined: 26 Feb 2011
Posts: 29
Re: If positive integers x and y are not both odd, which of the following  [#permalink]

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02 Mar 2011, 05:06
Yeah, exactly ... I was asking for x+y and x-y separately ... my fault ... however both explanations are useful
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Re: If positive integers x and y are not both odd, which of the following  [#permalink]

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15 May 2016, 04:03
If positive integers x and y are not both odd, which of the following must be even?

(A) xy
(B) x + y
(C) x - y
(D) x + y -1
(E) 2(x + y) - 1

Positive integers x and y are NOT both odd, means that either both x and y are even or one is even and the other one is odd. In either case xy must be even.

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-positive-integers-x-and-y-are-not-both-odd-which-of-the-165827.html
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Re: If positive integers x and y are not both odd, which of the following   [#permalink] 15 May 2016, 04:03
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