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

It is currently 16 Jan 2019, 12:31

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
Events & Promotions in January
PrevNext
SuMoTuWeThFrSa
303112345
6789101112
13141516171819
20212223242526
272829303112
Open Detailed Calendar
  • The winning strategy for a high GRE score

     January 17, 2019

     January 17, 2019

     08:00 AM PST

     09:00 AM PST

    Learn the winning strategy for a high GRE score — what do people who reach a high score do differently? We're going to share insights, tips and strategies from data we've collected from over 50,000 students who used examPAL.
  • Free GMAT Strategy Webinar

     January 19, 2019

     January 19, 2019

     07:00 AM PST

     09:00 AM PST

    Aiming to score 760+? Attend this FREE session to learn how to Define your GMAT Strategy, Create your Study Plan and Master the Core Skills to excel on the GMAT.

If x and y are nonzero integers, is x^y < y^x ?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

 
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 52161
If x and y are nonzero integers, is x^y < y^x ?  [#permalink]

Show Tags

New post 03 Mar 2014, 23:14
4
55
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

67% (01:42) correct 33% (01:47) wrong based on 1361 sessions

HideShow timer Statistics

The Official Guide For GMAT® Quantitative Review, 2ND Edition

If x and y are nonzero integers, is x^y < y^x ?

(1) x = y^2
(2) y > 2

Data Sufficiency
Question: 121
Category: Arithmetic; Algebra Arithmetic operations; Inequalities
Page: 161
Difficulty: 650

_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 52161
Re: If x and y are nonzero integers, is x^y < y^x ?  [#permalink]

Show Tags

New post 03 Mar 2014, 23:14
9
11
SOLUTION

If x and y are nonzero integers, is x^y < y^x ?

(1) x = y^2 --> if \(x=y=1\), then \(x^y=1=y^x\), so the answer would be NO BUT if \(y=3\) and \(x=9\), then \(x^y=9^3<y^x=3^9\), so the answer would be YES. Not sufficient.

(2) y > 2. No info about \(x\), not sufficient.

(1)+(2) From (1) \(x = y^2\), thus the question becomes: is \((y^2)^y<y^{(y^2)}\)? --> is \(y^{2y}<y^{(y^2)}\)? Now, since from (2) \(y=integer>2\), then \(2y\) will always be less than \(y^2\), therefore \(y^{2y}\) will be less than \(y^{(y^2)}\). Sufficient.

Answer: C.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

General Discussion
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 52161
Re: If x and y are nonzero integers, is x^y < y^x ?  [#permalink]

Show Tags

New post 08 Mar 2014, 12:51
2
1
SOLUTION

If x and y are nonzero integers, is x^y < y^x ?

(1) x = y^2 --> if \(x=y=1\), then \(x^y=1=y^x\), so the answer would be NO BUT if \(y=3\) and \(x=9\), then \(x^y=9^3<y^x=3^9\), so the answer would be YES. Not sufficient.

(2) y > 2. No info about \(x\), not sufficient.

(1)+(2) From (1) \(x = y^2\), thus the question becomes: is \((y^2)^y<y^{(y^2)}\)? --> is \(y^{2y}<y^{(y^2)}\)? Now, since from (2) \(y=integer>2\), then \(2y\) will always be less than \(y^2\), therefore \(y^{2y}\) will be less than \(y^{(y^2)}\). Sufficient.

Answer: C.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Math Revolution GMAT Instructor
User avatar
V
Joined: 16 Aug 2015
Posts: 6804
GMAT 1: 760 Q51 V42
GPA: 3.82
Premium Member
Re: If x and y are nonzero integers, is x^y < y^x ?  [#permalink]

Show Tags

New post 27 Jan 2018, 20:40
Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 2 variables (\(x\) and \(y\)) and 0 equations, C is most likely to be the answer. So, we should consider 1) & 2) first.

Conditions 1) & 2):
\(x^y = (y^2)^y = y^{2y}\)
\(y^x = y^{y^2}\)
Since \(y > 2\), we have \(y^2 > 2y\) and \(y^{2y} < y^{y^2}\).
Thus both conditions together are sufficient.

Since this is an integer question (one of the key question areas), we should also consider choices A and B by CMT(Common Mistake Type) 4(A).

Condition 1)
\(x = 9, y = 3 ⇒ 9^3 = 3^6 < 3^9\) : Yes
\(x = 1, y = 1 ⇒ 1^1 = 1^1\) : No
The condition 1) only is not sufficient.

Condition 2)
\(x = 1, y = 4 ⇒ 1^2 < 4^1\) : Yes
\(x = -1, y = 4 => (-1)^4 = 1 > 4^{-1} = \frac{1}{4}\) : No
The condition 2) only is not sufficient.

Therefore, C is the answer.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
_________________

MathRevolution: Finish GMAT Quant Section with 10 minutes to spare
The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy.
"Only $149 for 3 month Online Course"
"Free Resources-30 day online access & Diagnostic Test"
"Unlimited Access to over 120 free video lessons - try it yourself"

CEO
CEO
User avatar
D
Joined: 11 Sep 2015
Posts: 3319
Location: Canada
Re: If x and y are nonzero integers, is x^y < y^x ?  [#permalink]

Show Tags

New post 09 Oct 2018, 05:46
1
Top Contributor
Bunuel wrote:
If x and y are nonzero integers, is x^y < y^x ?

(1) x = y²
(2) y > 2


Target question: Is x^y < y^x ?

Given: x and y are nonzero integers

Statement 1: x = y²
Take the original target question, and replace x with y² to get: Is (y²)^y < y^(y²)?
Simplify the left side power to get: Is y^(2y) < y^(y²)?

Let's TEST some values of y.
Case a: If y = 1, then y^(2y) = 1 and y^(y²) = 1. In this case, the answer to the target question is NO, it is NOT the case that y^(2y) < y^(y²)
Case b: If y = 3, then y^(2y) = 3^6 and y^(y²) = 3^9. In this case, the answer to the target question is YES, it IS the case that y^(2y) < y^(y²)
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: y > 2
Since we have no information about x, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
From statement 1, we changed the target question to read Is y^(2y) < y^(y²)?
Now let's divide both sides of the inequality by y^(2y) to get: Is 1 < [y^(y²)/y^(2y)]?
Apply the Quotient Law to get: Is 1 < y^(y² - 2y)?
Notice that, if y is an INTEGER greater than 2 (i.e., statement 2), then y² - 2y is greater than 1
So, y^(y² - 2y) MUST be greater than 1
So, the answer to the target question is YES it IS the case that 1 < y^(y² - 2y)
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer: C

Cheers,
Brent
_________________

Test confidently with gmatprepnow.com

Manager
Manager
User avatar
S
Status: On the journey of achieving
Affiliations: Senior Manager, CA by profession, CFA(USA) Level 2
Joined: 06 Feb 2016
Posts: 203
Location: India
Concentration: Finance, Finance
GMAT 1: 560 Q44 V21
GPA: 3.82
WE: Other (Commercial Banking)
Reviews Badge CAT Tests
Re: If x and y are nonzero integers, is x^y < y^x ?  [#permalink]

Show Tags

New post 16 Oct 2018, 19:15
1
From Statement 1 x=y^2 so x and y both can be equal to 1 , then x^y cannot be less than y^x as both are equal
When y=3 then x=9 then x^y will be less than y^x. So no unique answer. Hence statement 1 alone is not suffice

Statement 2, y>2 does not provides us with any data about x hence not suffice as we cannot compare the expressions

Both statements together are required. When we combine both the statements, only value possible is y=3 and x=9
Hence Option C

Hence answer is Option C
_________________

Never Settle for something less than what you deserve...........

I've failed over and over and over again in my life and that is why I succeed--Michael Jordan
Kudos drives a person to better himself every single time. So Pls give it generously
Wont give up till i hit a 700+

GMAT Club Bot
Re: If x and y are nonzero integers, is x^y < y^x ? &nbs [#permalink] 16 Oct 2018, 19:15
Display posts from previous: Sort by

If x and y are nonzero integers, is x^y < y^x ?

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


Copyright

GMAT Club MBA Forum Home| About| Terms and Conditions and Privacy Policy| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.