GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 07 Dec 2019, 04:50 ### 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.  # For integers x and y, which of the following MUST be an integer?

Author Message
TAGS:

### Hide Tags

Math Expert V
Joined: 02 Sep 2009
Posts: 59589
For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

1
14 00:00

Difficulty:   55% (hard)

Question Stats: 62% (01:32) correct 38% (01:33) wrong based on 315 sessions

### HideShow timer Statistics

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. $$\sqrt{81x^2+25xy+16y^2}$$

_________________
GMAT Tutor G
Joined: 24 Jun 2008
Posts: 1829
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

6
1
Smokeybear00 wrote:
Is there an approach to this if one does not immediately recognize (x^2 - 2xy +y^2) ?

You'll have more flexibility in general if you can recognize how to factor in questions like this, but as a fallback, an alternative approach is to plug in some simple numbers for x and y. The right answer needs to always give us an integer result, so if we find that an answer does not give an integer result for some numbers we pick, that cannot be the right answer. And if you plug in x=0 and y=1, you get a negative number under the square root for C and D, so you get something undefined (or 'imaginary' in advanced math language), so those can't be right answers. If you plug in x=1 and y=1, you get √91 for A, and √122 for E, which aren't integers either, so those can't be right. That only leaves D, which does give us an integer in both cases.

If you ever need to use an approach like this, try to focus on the simplest possible sets of numbers, so you don't spent too long doing calculations. Zero (when allowed) can be a very convenient choice, and sometimes, as in this question, just plugging in 1 for everything will get you the answer (we actually could have just plugged in x=1 and y=1 and that would have been enough here - only D gives an integer - but plugging in x=0 rules out two answers so quickly I prefer to do that). At the very least, you usually will rule out three wrong answers by using very simple numbers, and then will only need to plug slightly more complicated numbers into two answer choices instead of five.
_________________
GMAT Tutor in Toronto

If you are looking for online GMAT math tutoring, or if you are interested in buying my advanced Quant books and problem sets, please contact me at ianstewartgmat at gmail.com
##### General Discussion
Manager  S
Joined: 23 May 2017
Posts: 231
Concentration: Finance, Accounting
WE: Programming (Energy and Utilities)
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

Ans : B

B can be written as = (( 7x - 6y) ^2)^1/2 = 7x - 6y = integer
Senior Manager  S
Joined: 21 Mar 2016
Posts: 494
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

1
its a direct application of formula...
(a+b)^2 = a^2 + 2ab + b^2

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

ans B
Current Student S
Joined: 30 May 2017
Posts: 65
Concentration: Finance, General Management
GMAT 1: 690 Q47 V38 GPA: 3.23
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

Is there an approach to this if one does not immediately recognize (x^2 - 2xy +y^2) ?
_________________
Veritas Prep 6/18/17 600 Q:38 V:35 IR:5
Veritas Prep 6/29/17 620 Q:43 V:33 IR:4
Manhattan 7/12/17 640 Q:42 V:35 IR:2.4
Veritas Prep 7/27/17 640 Q:41 V:37 IR:4
Manhattan 8/9/17 670 Q:44 V:37 IR:3
Veritas Prep 8/21/17 660 Q:45 V:36 IR:7
GMAT Prep 8/23/17 700 Q:47 V:38 IR:8
GMAT Prep 8/27/17 730 Q:49 V:40 IR:8
Veritas Prep 8/30/17 690 Q:47 V:37 IR:8
SVP  B
Joined: 06 Nov 2014
Posts: 1870
For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

1
1
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
Director  G
Joined: 02 Sep 2016
Posts: 641
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

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
Target Test Prep Representative V
Status: Founder & CEO
Affiliations: Target Test Prep
Joined: 14 Oct 2015
Posts: 8622
Location: United States (CA)
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

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.

_________________

# Scott Woodbury-Stewart

Founder and CEO

Scott@TargetTestPrep.com

See why Target Test Prep is the top rated GMAT quant course on GMAT Club. Read Our Reviews

If you find one of my posts helpful, please take a moment to click on the "Kudos" button.

Manager  B
Joined: 30 Apr 2013
Posts: 75
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

Can we not apply the same rule to options A and D?
Manager  B
Joined: 30 Apr 2013
Posts: 75
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

I mean option A and E
Senior SC Moderator V
Joined: 22 May 2016
Posts: 3723
For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

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.
_________________
SC Butler has resumed! Get two SC questions to practice, whose links you can find by date, here.

Never doubt that a small group of thoughtful, committed citizens can change the world; indeed, it's the only thing that ever has -- Margaret Mead
GMAT Club Legend  V
Joined: 12 Sep 2015
Posts: 4125
For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

1
Top Contributor
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. $$\sqrt{81x^2+25xy+16y^2}$$

Another approach:

The question is asking us to determine which expression MUST be an integer for ALL integer values of x and y.
So, let's TEST a pair of values.
Let's plug in x = 1 and y = 1
If an expression evaluates to be a non-integer, we can ELIMINATE that answer choice.

We get...
A)√91.This does NOT evaluate to be an integer. ELIMINATE C
B)√1 = 1. This IS an integer. So, keep B
C)√15. This does NOT evaluate to be an integer. ELIMINATE C
D)√-64. Cannot evaluate. ELIMINATE D
E)√122. This does NOT evaluate to be an integer. ELIMINATE E

By the process of elimination, the correct answer is B

Cheers,
Brent
_________________

Originally posted by GMATPrepNow on 13 Sep 2017, 15:34.
Last edited by GMATPrepNow on 06 May 2019, 20:27, edited 2 times in total.
Non-Human User Joined: 09 Sep 2013
Posts: 13724
Re: For integers x and y, which of the following MUST be an integer?  [#permalink]

### Show Tags

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: For integers x and y, which of the following MUST be an integer?   [#permalink] 08 Oct 2019, 10:13
Display posts from previous: Sort by

# For integers x and y, which of the following MUST be an integer?  