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

Question

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

ScottTargetTestPrep · Accepted Answer

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.

Answer: D

rajatbhatnagar91 · Answer

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.
Answer - D

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Ejiroosa · Answer

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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sashiim20 · Answer

(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)...

ydmuley · Answer

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

adkikani · Answer

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?

amanvermagmat · Answer

Hello

yes i think this approach is correct

GMATinsight · Answer

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

GMATinsight · Answer

Answer: Option D

Video solution by  GMATinsight

https://www.youtube.com/watch?v=dOjGBes8Kyo

BrentGMATPrepNow · Answer

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,
Brent

Sneha2021 · Answer

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!

BrentGMATPrepNow · Answer

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).

KarishmaB · Answer

'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.

avigutman · Answer

Video solution from Quant Reasoning: Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1 https://www.youtube.com/watch?v=pasVtNR8fHE

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

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