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If x and y are positive integers, is xy even? : Data Sufficiency (DS)
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Re: If x and y are positive integers, is xy even? [#permalink]
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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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If x and y are positive integers, is xy even? [#permalink]
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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.

Answer (D)...
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Re: If x and y are positive integers, is xy even? [#permalink]
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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

Hence, the answer is D
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If x and y are positive integers, is xy even? [#permalink]
Bunuel pushpitkc niks18 Hatakekakashi
amanvermagmat

How about this approach?

\(x^2\) + \(y^2\) - 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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adkikani wrote:
Bunuel pushpitkc niks18 Hatakekakashi
amanvermagmat

How about this approach?

\(x^2\) + \(y^2\) - 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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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.


Question: Is x*y even?

STatement 1: x^2 + y^2 − 1 = 4a

i.e. x^2 + y^2 = 4a + 1 = ODD
i.e. one of x and y must be even and other must be odd

i.e. x*y = even

SUFFICIENT

Statement 2: x + y = odd

i.e. one of x and y must be even and other must be odd

SUFFICIENT

ANswer: Option D
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Re: If x and y are positive integers, is xy even? [#permalink]
Expert Reply
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.


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Answer: Option D

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

(1) x² + y² − 1 is divisible by 4.
(2) x + y is odd.


Some important rules:
#1. ODD +/- ODD = EVEN
#2. ODD +/- EVEN = ODD
#3. EVEN +/- EVEN = EVEN

#4. (ODD)(ODD) = ODD
#5. (ODD)(EVEN) = EVEN
#6. (EVEN)(EVEN) = EVEN


Target question: Is xy even?

Given: x and y are positive integers

Statement 1: x² + y² − 1 is divisible by 4.
In other words, x² + y² − 1 is EVEN
This means x² + y² is ODD.
If x² + y² is ODD, then one of the values (x² or y²) is ODD, and the other value (x² or y²) is EVEN
If one of the values (x² or y²) is ODD, then the individual value (x or y) is ODD.
If the other value (x² or y²) is EVEN, then that other value is EVEN.
So, one value (x or y) is ODD, and the other value is EVEN, which means the product xy is EVEN.
Statement 1 is SUFFICIENT

Statement 2: x + y is odd.
This means one value (x or y) is ODD, and the other value is EVEN, which means the product xy is EVEN.
Statement 2 is SUFFICIENT

Answer: D

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

What is the significance of x and y are positive integers?
If the question says x and y are integers, answer will still be D.

Thank you for your time!
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Re: If x and y are positive integers, is xy even? [#permalink]
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Sneha2021 wrote:
BrentGMATPrepNow Bunuel KarishmaB

What is the significance of x and y are positive integers?
If the question says x and y are integers, answer will still be D.

Thank you for your time!


Yes, if the question just said that x and y are integers, the correct answer would still be D.
The test-makers typically add the "positive integers" to integer properties questions to avoid students having to deal with questions like "Is 0 a multiple of 5?" (answer: yes).
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Re: If x and y are positive integers, is xy even? [#permalink]
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Sneha2021 wrote:
BrentGMATPrepNow Bunuel KarishmaB

What is the significance of x and y are positive integers?
If the question says x and y are integers, answer will still be D.

Thank you for your time!


'a is divisible by b' is typically a concept of positive integers. Factors are positive integers. Hence, it makes sense for them to specify that x and y are positive integers even if it doesn't matter in the question.
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Re: If x and y are positive integers, is xy even? [#permalink]
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Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
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