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# If x and y are positive integers, is xy even?

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If x and y are positive integers, is xy even?  [#permalink]

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26 Jun 2017, 02:43
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If x and y are positive integers, is xy even?

(1) x^2 + y^2 − 1 is divisible by 4.
(2) x + y is odd.

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Re: If x and y are positive integers, is xy even?  [#permalink]

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Updated on: 25 Jul 2017, 04:36
1
Statement 1
X^2 +Y^2 - 1 is divisible by 4. can be 3^2 + 5^2 - 1 = 24 or 4^2 + 1^2 - 1 = 16 S
Statement 2. X+Y = odd. it has to be one even and one odd number. Suff

Ans D

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Originally posted by Ejiroosa on 26 Jun 2017, 02:50.
Last edited by Ejiroosa on 25 Jul 2017, 04:36, edited 1 time in total.
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Re: If x and y are positive integers, is xy even?  [#permalink]

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26 Jun 2017, 02:54
4
x^2 + y^2 - 1 has to be even to be divisible by 4.
Hence x^2 + y^2 is odd.
This means either x or y has to be even. Statement 1 is sufficient.
Similarly for x+y to be odd, either x or y has to be even. Hence product xy is even.
Statement 2 is also sufficient.

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If x and y are positive integers, is xy even?  [#permalink]

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26 Jun 2017, 02:59
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Bunuel wrote:
If x and y are positive integers, is xy even?

(1) x^2 + y^2 − 1 is divisible by 4.
(2) x + y is odd.

(1) $$x^2 + y^2 − 1$$ is divisible by 4.

Let ($$x^2 + y^2$$ ) be $$z$$.

$$z - 1$$ is divisible by $$4$$.

Only even number is divisible by $$4$$. Hence $$z - 1$$ should be even.

Odd - Odd = Even.

Therefore $$(x^2 - y^2)$$ should be Odd. ($$Odd^2$$ will be Odd. $$Even^2$$ will be Even)

Even - Odd = Odd

Therefore either $$x$$ or $$y$$ should be even. Therefore $$xy$$ will be even. I is Sufficient.

(2) $$x + y$$ is odd.

Even + Odd = Odd.

Therefore either $$x$$ or $$y$$ should be even. Therefore $$xy$$ will be even. II is Sufficient.

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Re: If x and y are positive integers, is xy even?  [#permalink]

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26 Jun 2017, 03:07
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If x and y are positive integers, is $$xy$$even?

(1) $$x^2 + y^2 − 1$$ is divisible by 4

This means that either x or y has to be odd.

As you know ODD * EVEN = EVEN

Question - Is xy EVEN ? is TRUE =====> Eq. (1) SUFFICIENT

(2) $$x + y$$ is odd

As we know, ODD + EVEN = ODD

And ODD * EVEN = EVEN

Question - Is xy EVEN ? is TRUE =====> Eq. (2) SUFFICIENT

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Re: If x and y are positive integers, is xy even?  [#permalink]

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26 Jun 2017, 04:30
Ans is D used 5 instead of 4....

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Re: If x and y are positive integers, is xy even?  [#permalink]

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17 Jul 2017, 22:45
Bunuel wrote:
If x and y are positive integers, is xy even?

(1) x^2 + y^2 − 1 is divisible by 4.
(2) x + y is odd.

St 1

(x^2 + y^2 − 1) /4 = some integer - therefore some odd number minus 1 is divisible by 4

x^2 + y^2= some odd number... in order for this to be true either X and Y must be some even and odd mix

(1)^2 +(2)^2= 5 odd

knowing x and y must be different ( even and odd) x and y must odd

St 2

Even + Odd = Odd so

Suff

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Re: If x and y are positive integers, is xy even?  [#permalink]

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30 Jul 2017, 17:31
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Bunuel wrote:
If x and y are positive integers, is xy even?

(1) x^2 + y^2 − 1 is divisible by 4.
(2) x + y is odd.

We need to determine whether the product of x and y is even.

Statement One Alone:

x^2 + y^2 − 1 is divisible by 4.

Since 4 is an even number, we need x^2 + y^2 − 1 to be even. In order for x^2 + y^2 − 1 to be even, we need x^2 + y^2 to be odd. If the sum of two squares is odd, one of them must be odd and the other must be even. This means that either x = odd and y = even OR x = even and y = odd. In either case, the product of x and y will be even. Statement one alone is sufficient to answer the question.

Statement Two Alone:

x + y is odd.

Since x + y = odd, either x = odd and y = even OR x = even and y = odd. In either case, the product of x and y will be even. Statement two alone is sufficient to answer the question.

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If x and y are positive integers, is xy even?  [#permalink]

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13 Mar 2018, 21:18
Bunuel pushpitkc niks18 Hatakekakashi
amanvermagmat

$$x^2$$ + $$y^2$$ - 1 = 4 *(m) where m is an integer since there is no remainder (given)

or

$$x^2$$ + $$y^2$$ = odd

or

x + y = odd (a positive odd / even no when squared will give odd and even values respectively)

This is only possible when either of x or y is odd.

Does that help to make St 1 suff?
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Re: If x and y are positive integers, is xy even?  [#permalink]

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13 Mar 2018, 21:54
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Bunuel pushpitkc niks18 Hatakekakashi
amanvermagmat

$$x^2$$ + $$y^2$$ - 1 = 4 *(m) where m is an integer since there is no remainder (given)

or

$$x^2$$ + $$y^2$$ = odd

or

x + y = odd (a positive odd / even no when squared will give odd and even values respectively)

This is only possible when either of x or y is odd.

Does that help to make St 1 suff?

Hello

yes i think this approach is correct
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Re: If x and y are positive integers, is xy even?  [#permalink]

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14 Mar 2018, 00:21
Bunuel pushpitkc niks18 Hatakekakashi
amanvermagmat

$$x^2$$ + $$y^2$$ - 1 = 4 *(m) where m is an integer since there is no remainder (given)

or

$$x^2$$ + $$y^2$$ = odd

or

x + y = odd (a positive odd / even no when squared will give odd and even values respectively)

This is only possible when either of x or y is odd.

Does that help to make St 1 suff?

this approach is correct

good usage of basic concepts (Y)
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Re: If x and y are positive integers, is xy even?  [#permalink]

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19 Mar 2019, 08:19
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Re: If x and y are positive integers, is xy even?   [#permalink] 19 Mar 2019, 08:19
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