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

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
_________________

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

Sent from my ONEPLUS A3003 using GMAT Club Forum mobile app

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

Sent from my SM-G935F using GMAT Club Forum mobile app

(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)...

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
_________________

Ans is D used 5 instead of 4....

Sent from my SM-G935F using GMAT Club Forum mobile app

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

D

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

this approach is correct

good usage of basic concepts (Y)
_________________

1) x^2 + y^2 is odd. So, one of x and y will be even which will make xy even Sufficient
2) same as 1. Sufficient.
D is the 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

****************
x,y>0 , is xy even?

xy can be even when

x and y both are even or
either x or y is even

(1) x^2 + y^2 − 1 is divisible by 4, therefore, x^2 + y^2 − 1=4I (I is an integer)

x^2 + y^2 =4I+ 1 (can you correlate it with 2n+1) for x^2 + y^2 to be odd, both x and y should be odd.
Odd*Odd=Odd
Sufficient.

(2) x + y is odd (Refer above).

Sufficient.

Answer is D

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

Two scenarios:
even x even
even x odd

(1) x^2 + y^2 − 1 is divisible by 4.

For x^2 + y^2 − 1 to be divisible by 4, we must have a multiple of 4 in the numerator (i.e. the numerator has to be even).

even + odd - odd = even
odd + even - odd = even

Sufficient

(2) x + y is odd.

odd + even = odd
even + odd = odd

Sufficient
_________________

Answer: Option D

Video solution by GMATinsight



Get TOPICWISE: Concept Videos | Practice Qns 100+ | Official Qns 50+ | 100% Video solution CLICK HERE.
Two MUST join YouTube channels : GMATinsight (1000+ FREE Videos) and GMATclub

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

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).

'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.
_________________