[phpBB Debug] PHP Notice: in file /includes/check_new_recommended_questions.php on line 37: Undefined array key "last_recommended_questions_epoch"
[phpBB Debug] PHP Notice: in file /includes/check_new_recommended_questions.php on line 41: Undefined array key "last_recommended_questions_epoch"
If x and y are positive integers, is xy even? : Data Sufficiency (DS)
 Last visit was: 20 Jul 2024, 02:32 It is currently 20 Jul 2024, 02:32
Toolkit
GMAT Club Daily Prep
Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History
Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

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

SORT BY:
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94422
Own Kudos [?]: 642445 [115]
Given Kudos: 86334
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19175
Own Kudos [?]: 22679 [36]
Given Kudos: 286
Location: United States (CA)
Intern
Joined: 26 Feb 2017
Posts: 3
Own Kudos [?]: 7 [7]
Given Kudos: 1
General Discussion
Intern
Joined: 27 May 2017
Posts: 10
Own Kudos [?]: 9 [1]
Given Kudos: 0
Re: If x and y are positive integers, is xy even? [#permalink]
1
Kudos
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

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.
Director
Joined: 04 Dec 2015
Posts: 620
Own Kudos [?]: 1616 [5]
Given Kudos: 276
Location: India
Concentration: Technology, Strategy
WE:Information Technology (Consulting)
If x and y are positive integers, is xy even? [#permalink]
3
Kudos
2
Bookmarks
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.

Retired Moderator
Joined: 19 Mar 2014
Posts: 815
Own Kudos [?]: 981 [6]
Given Kudos: 199
Location: India
Concentration: Finance, Entrepreneurship
GPA: 3.5
Re: If x and y are positive integers, is xy even? [#permalink]
5
Kudos
1
Bookmarks
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

IIM School Moderator
Joined: 04 Sep 2016
Posts: 1253
Own Kudos [?]: 1265 [0]
Given Kudos: 1207
Location: India
WE:Engineering (Other)
If x and y are positive integers, is xy even? [#permalink]
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?
Retired Moderator
Joined: 22 Aug 2013
Posts: 1181
Own Kudos [?]: 2552 [1]
Given Kudos: 459
Location: India
Re: If x and y are positive integers, is xy even? [#permalink]
1
Kudos
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
GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6020
Own Kudos [?]: 13810 [2]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Re: If x and y are positive integers, is xy even? [#permalink]
1
Kudos
1
Bookmarks
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

GMAT Club Legend
Joined: 08 Jul 2010
Status:GMAT/GRE Tutor l Admission Consultant l On-Demand Course creator
Posts: 6020
Own Kudos [?]: 13810 [0]
Given Kudos: 125
Location: India
GMAT: QUANT+DI EXPERT
Schools: IIM (A) ISB '24
GMAT 1: 750 Q51 V41
WE:Education (Education)
Re: If x and y are positive integers, is xy even? [#permalink]
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.

Wanna make solving the Official Questions interesting???

Click here and solve 1000+ Official Questions with Video solutions as Timed Sectional Tests
and Dedicated Data Sufficiency (DS) Course

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
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6804
Own Kudos [?]: 30839 [2]
Given Kudos: 799
Re: If x and y are positive integers, is xy even? [#permalink]
1
Kudos
1
Bookmarks
Top Contributor
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

Cheers,
Brent
Senior Manager
Joined: 20 Dec 2020
Posts: 286
Own Kudos [?]: 30 [0]
Given Kudos: 497
Location: India
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.

GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6804
Own Kudos [?]: 30839 [0]
Given Kudos: 799
Re: If x and y are positive integers, is xy even? [#permalink]
Top Contributor
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.

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).
Tutor
Joined: 16 Oct 2010
Posts: 15125
Own Kudos [?]: 66741 [0]
Given Kudos: 436
Location: Pune, India
Re: If x and y are positive integers, is xy even? [#permalink]
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.

'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.
Tutor
Joined: 17 Jul 2019
Posts: 1304
Own Kudos [?]: 1745 [0]
Given Kudos: 66
GMAT 1: 780 Q51 V45
GMAT 2: 780 Q50 V47
GMAT 3: 770 Q50 V45
Re: If x and y are positive integers, is xy even? [#permalink]
Video solution from Quant Reasoning:
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1
Non-Human User
Joined: 09 Sep 2013
Posts: 34042
Own Kudos [?]: 853 [0]
Given Kudos: 0
Re: If x and y are positive integers, is xy even? [#permalink]
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
Re: If x and y are positive integers, is xy even? [#permalink]
Moderator:
Math Expert
94421 posts