Last visit was: 13 Jul 2024, 18:09 It is currently 13 Jul 2024, 18:09
Close
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
Your Progress

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.
Close
Request Expert Reply
Confirm Cancel
SORT BY:
Kudos
Tags:
Show Tags
Hide Tags
Math Expert
Joined: 02 Sep 2009
Posts: 94341
Own Kudos [?]: 640537 [31]
Given Kudos: 85005
Send PM
Most Helpful Reply
GMAT Tutor
Joined: 24 Jun 2008
Posts: 4128
Own Kudos [?]: 9437 [10]
Given Kudos: 91
 Q51  V47
Send PM
GMAT Club Legend
GMAT Club Legend
Joined: 12 Sep 2015
Posts: 6805
Own Kudos [?]: 30793 [5]
Given Kudos: 799
Location: Canada
Send PM
General Discussion
SVP
SVP
Joined: 06 Nov 2014
Posts: 1793
Own Kudos [?]: 1378 [3]
Given Kudos: 23
Send PM
For integers x and y, which of the following MUST be an integer? [#permalink]
1
Kudos
2
Bookmarks
Expert Reply
Bunuel wrote:
For integers x and y, which of the following MUST be an integer?


A. \(\sqrt{25x^2+30xy+36y^2}\)

B. \(\sqrt{49x^2−84xy+36y^2}\)

C. \(\sqrt{16x^2−y^2}\)

D. \(\sqrt{64x^2−64xy−64y^2}\)

E. \(\sqrt{81x^2+25xy+16y^2}\)


If the term inside the square root is a perfect square, then the whole term will be an integer.
We need to try and write the terms in the for of x^2 +- 2ax + a^2

A. \(\sqrt{25x^2+30xy+36y^2}\) = \(\sqrt{25x^2+5*6 xy+36y^2}\) this cannot be a perfect square

B. \(\sqrt{49x^2−84xy+36y^2}\) = \(\sqrt{49x^2−2*6*7*xy+36y^2}\) = \(\sqrt{(7x−6y)^2}\)
This is a perfect square.

Correct Option: B
Target Test Prep Representative
Joined: 14 Oct 2015
Status:Founder & CEO
Affiliations: Target Test Prep
Posts: 19130
Own Kudos [?]: 22627 [2]
Given Kudos: 286
Location: United States (CA)
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
2
Kudos
Expert Reply
Bunuel wrote:
For integers x and y, which of the following MUST be an integer?


A. \(\sqrt{25x^2+30xy+36y^2}\)

B. \(\sqrt{49x^2−84xy+36y^2}\)

C. \(\sqrt{16x^2−y^2}\)

D. \(\sqrt{64x^2−64xy−64y^2}\)

E. \(\sqrt{81x^2+25xy+16y^2}\)


We need to determine which of the answer choices must be an integer.

Let’s take a look at answer choice B:

√(49x^2 - 84xy + 36y^2)

√(7x - 6y)(7x - 6y)

√(7x - 6y)^2 = |7x - 6y|

Since x and y are integers, |7x - 6y| is an integer.

Answer: B
Senior Manager
Senior Manager
Joined: 21 Mar 2016
Posts: 448
Own Kudos [?]: 120 [1]
Given Kudos: 103
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
1
Bookmarks
its a direct application of formula...
(a+b)^2 = a^2 + 2ab + b^2

(a-b)^2 = a^2 -2ab + b^2

ans B
Senior SC Moderator
Joined: 22 May 2016
Posts: 5327
Own Kudos [?]: 35753 [1]
Given Kudos: 9464
Send PM
For integers x and y, which of the following MUST be an integer? [#permalink]
1
Kudos
Expert Reply
Bunuel wrote:
For integers x and y, which of the following MUST be an integer?

A. \(\sqrt{25x^2+30xy+36y^2}\)

B. \(\sqrt{49x^2−84xy+36y^2}\)

C. \(\sqrt{16x^2−y^2}\)

D. \(\sqrt{64x^2−64xy−64y^2}\)

E. \(\sqrt{81x^2+25xy+16y^2}\)

santro789 wrote:
Can we not apply the same rule to options A and D E?

santro789 , I'm not sure which rule you mean. In fact, I'm slightly confused by your question. Under either rule, how do you see A and E as possible answers?

OptimusPrepJanielle wrote
Quote:
If the term inside the square root is a perfect square, then the whole term will be an integer.
We need to try and write the terms in the form of x^2 +- 2ax + a^2

ScottTargetTestPrep wrote
Quote:
√(7x - 6y)^2 = |7x - 6y|

Since x and y are integers, |7x - 6y| is an integer.

The answer to your question is no. The most basic reason: the middle term in both A and E prevents both from being perfect squares.

If A were a perfect square, looking at its terms' coefficients, it would be
\(\sqrt{(5x + 6y)^2}\)= \(\sqrt{25x^2 + 60xy + 36y^2}\)

Answer A's middle term is 30xy, not 60xy. That means it's not a perfect square.

Answer E has the same problem. Looking at its coefficients, if it were a perfect square it would be
\(\sqrt{(9x + 4y)^2}\\
=\sqrt{81x^2 + 72xy + 16y^2}\)

The middle term in E is 25xy, not 72xy. In neither A nor E can we get a perfect square under the square root sign. If we could, we would get an integer: the square root of a perfect square is an integer. Just think about a couple of numeric values: \(\sqrt{2^2} = 2\), and \(\sqrt{41^2} = 41\)

Without being able to factor A and E into \((a + b)^2\) or \((a - b)^2\) because their middle terms prevent them from being perfect squares, we certainly cannot use absolute value analysis to prove what they are not.

Hope that helps.
Manager
Manager
Joined: 23 May 2017
Posts: 191
Own Kudos [?]: 360 [0]
Given Kudos: 9
Concentration: Finance, Accounting
WE:Programming (Energy and Utilities)
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
Ans : B

B can be written as = (( 7x - 6y) ^2)^1/2 = 7x - 6y = integer
Manager
Manager
Joined: 30 May 2017
Posts: 57
Own Kudos [?]: 62 [0]
Given Kudos: 42
Concentration: Finance, General Management
GMAT 1: 690 Q47 V38
GPA: 3.23
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
Is there an approach to this if one does not immediately recognize (x^2 - 2xy +y^2) ?
Director
Director
Joined: 02 Sep 2016
Posts: 525
Own Kudos [?]: 195 [0]
Given Kudos: 277
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
The options are the expanded form of identities which are commonly tested on GMAT.

B is the answer and the identity used in B is: (a-b)^2= a^2+b^2-2ab
Manager
Manager
Joined: 30 Apr 2013
Posts: 60
Own Kudos [?]: 10 [0]
Given Kudos: 9
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
Can we not apply the same rule to options A and D?
Manager
Manager
Joined: 30 Apr 2013
Posts: 60
Own Kudos [?]: 10 [0]
Given Kudos: 9
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
I mean option A and E
User avatar
Non-Human User
Joined: 09 Sep 2013
Posts: 33963
Own Kudos [?]: 851 [0]
Given Kudos: 0
Send PM
Re: For integers x and y, which of the following MUST be an integer? [#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.
GMAT Club Bot
Re: For integers x and y, which of the following MUST be an integer? [#permalink]
Moderator:
Math Expert
94341 posts