Author 
Message 
TAGS:

Hide Tags

Manager
Joined: 05 Jun 2009
Posts: 108

If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
10 Jan 2010, 00:55
5
This post received KUDOS
11
This post was BOOKMARKED
Question Stats:
74% (00:45) correct 26% (01:07) wrong based on 1041 sessions
HideShow timer Statistics
If x is a positive integer, is \(\sqrt{x}\) an integer? (1) \(\sqrt{4x}\) is an integer. (2) \(\sqrt{3x}\) is not an integer. the explanation says since sqrt (4x) is an integer ,it follows that 4x must be square of an integer and so x must be square of an integer and therefore sqrt (x) is an integer . i was trying to solve it this way sqrt(4x) =integer ,=> 2. sqrt (x) =integer => sqrt (x) =integer/2 =integer or non integer for example if 2 .sqrt (x) = 4 ,=> sqrt (x) =2 and so sqrt (x) is integer but if 2. sqrt (x) =3 ,=> sqrt (x) =3/2 and so sqrt (x) is non integer
friends please help me in pointing out whee i am going wrong.
Thanks in advance .. OPEN DISCUSSION OF THIS QUESTION IS HERE: https://gmatclub.com/forum/ifxisapo ... 65976.html
Official Answer and Stats are available only to registered users. Register/ Login.
Last edited by Bunuel on 25 Nov 2017, 00:05, edited 2 times in total.
Renamed the topic, edited the question and added OA.



Manager
Joined: 06 Jan 2010
Posts: 68

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
10 Jan 2010, 08:42
2
This post received KUDOS
if x is a positive integer ,is sqrt (x) an integer
(1) sqrt(4x) is an integer .
(2) sqrt (3x) is not an integer .
the explanation says since sqrt (4x) is an integer ,it follows that 4x must be square of an integer and so x must be square of an integer and therefore sqrt (x) is an integer . i was trying to solve it this way sqrt(4x) =integer ,=> 2. sqrt (x) =integer => sqrt (x) =integer/2 =integer or non integer for example if 2 .sqrt (x) = 4 ,=> sqrt (x) =2 and so sqrt (x) is integer but if 2. sqrt (x) =3 ,=> sqrt (x) =3/2 and so sqrt (x) is non integer yes. sqrt(x) is an integer.
how i worked it out : sqrt(4x) is an integer ==> 2 * sqrt(x) is an integer
in order that the product of 2 and sqrt(x) be an integer, sqrt(x) must either be 1) an integer 2) exactly half of an integer. i.e a number like 0.5, 1.5, 2.5 etc etc you worked that out as well. sqrt(x) = integer/2
we also know x is an integer. is there any integer whose square root is a half of an integer ? no!
therefore the only other alternative is sqrt(x) is a whole integer.
hope that helped



Manager
Joined: 05 Jun 2009
Posts: 108

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
11 Jan 2010, 06:50
Hi Janani ,
Thanks for the explanation ,do we have a rule like we can't have a sqrt (x)=integer /2, or its by observation ...



Manager
Joined: 17 Jan 2010
Posts: 143
Concentration: General Management, Strategy
GPA: 3.78
WE: Engineering (Manufacturing)

If x is a positive integer, is root(x) an integer? [#permalink]
Show Tags
20 Feb 2010, 14:49
2
This post received KUDOS
15
This post was BOOKMARKED
If x is a positive integer, is \(\sqrt{x}\) an integer? (1) \(\sqrt{4x}\) is an integer. (2) \(\sqrt{3x}\) is not an integer. This is the question from GMAT Quant Review. My logic to solve this question: \sqrt{4x}=2*\sqrt{x}, so \sqrt{x} can either be integer or not an integer (for example \sqrt{x}=2.5) and the 2*\sqrt{x} is still an integer. So Statement 1 is insufficient. \sqrt{3x}= \sqrt{3}*\sqrt{x}. As \sqrt{3} is not an integer, the \sqrt{x} can be either integer or non integer and the \sqrt{3}*\sqrt{x} will still be not integer. So Statement 2 is insufficient. S1 and S2 together is still insufficient as \sqrt{x}=2 and \sqrt{x}=2.5 both satisfy statements requirement. So I choose E as an answer. Is there a flaw in my reasoning? OG Quant review answer to this question is different from E. Please advice.



Senior Manager
Joined: 22 Dec 2009
Posts: 350

Re: Is OG Quant question answer wrong? [#permalink]
Show Tags
20 Feb 2010, 15:14
alexBLR wrote: This is the question from GMAT Quant Review: If x is a positive integer , is \sqrt{x} an integer? 1) \sqrt{4x} is an integer. 2) \sqrt{3x} is not an integer. My logic to solve this question: \sqrt{4x}=2*\sqrt{x}, so \sqrt{x} can either be integer or not an integer (for example \sqrt{x}=2.5) and the 2*\sqrt{x} is still an integer. So Statement 1 is insufficient. \sqrt{3x}= \sqrt{3}*\sqrt{x}. As \sqrt{3} is not an integer, the \sqrt{x} can be either integer or non integer and the \sqrt{3}*\sqrt{x} will still be not integer. So Statement 2 is insufficient. S1 and S2 together is still insufficient as \sqrt{x}=2 and \sqrt{x}=2.5 both satisfy statements requirement. So I choose E as an answer. Is there a flaw in my reasoning? OG Quant review answer to this question is different from E. Please advice. IMO D... Ques: if x is a positive integer, is \(\sqrt{x}\) an integer? S1: \(\sqrt{4x}\) is an integer > \(2* \sqrt{x}\) is an integer > \(\sqrt{x}\) has to be an integer.. as x is a positive integer and hence cannot be a fraction. Therefore SUFF S2: \(\sqrt{3x}\) is an integer > \(\sqrt{3}*\sqrt{x}\) > \(\sqrt{x}\) is not an integer as same could be a of a form of \(a\sqrt{3}\) where 'a' is a positive integer. Therefore SUFF
_________________
Cheers! JT........... If u like my post..... payback in Kudos!!
Do not post questions with OAPlease underline your SC questions while postingTry posting the explanation along with your answer choice For CR refer Powerscore CR BibleFor SC refer Manhattan SC Guide
~~Better Burn Out... Than Fade Away~~



Math Expert
Joined: 02 Sep 2009
Posts: 44388

If x is a positive integer, is root(x) an integer? [#permalink]
Show Tags
20 Feb 2010, 15:57
5
This post received KUDOS
Expert's post
1
This post was BOOKMARKED
jeeteshsingh wrote: alexBLR wrote: This is the question from GMAT Quant Review: If x is a positive integer , is \sqrt{x} an integer? 1) \sqrt{4x} is an integer. 2) \sqrt{3x} is not an integer. My logic to solve this question: \sqrt{4x}=2*\sqrt{x}, so \sqrt{x} can either be integer or not an integer (for example \sqrt{x}=2.5) and the 2*\sqrt{x} is still an integer. So Statement 1 is insufficient. \sqrt{3x}= \sqrt{3}*\sqrt{x}. As \sqrt{3} is not an integer, the \sqrt{x} can be either integer or non integer and the \sqrt{3}*\sqrt{x} will still be not integer. So Statement 2 is insufficient. S1 and S2 together is still insufficient as \sqrt{x}=2 and \sqrt{x}=2.5 both satisfy statements requirement. So I choose E as an answer. Is there a flaw in my reasoning? OG Quant review answer to this question is different from E. Please advice. IMO D... Ques: if x is a positive integer, is \(\sqrt{x}\) an integer? S1: \(\sqrt{4x}\) is an integer > \(2* \sqrt{x}\) is an integer > \(\sqrt{x}\) has to be an integer.. as x is a positive integer and hence cannot be a fraction. Therefore SUFF S2: \(\sqrt{3x}\) is an integer > \(\sqrt{3}*\sqrt{x}\) > \(\sqrt{x}\) is not an integer as same could be a of a form of \(a\sqrt{3}\) where 'a' is a positive integer. Therefore SUFF If x is a positive integer, is \(\sqrt{x}\) an integer?As given that \(x\) is a positive integer then \(\sqrt{x}\) is either an integer itself or an irrational number. (1) \(\sqrt{4x}\) is an integer > \(2\sqrt{x}=integer\) > \(2\sqrt{x}\) to be an integer \(\sqrt{x}\) must be an integer or integer/2, but as \(x\) is an integer, then \(\sqrt{x}\) can not be integer/2, hence \(\sqrt{x}\) is an integer. Sufficient. (2) \(\sqrt{3x}\) is not an integer > if \(x=9\), condition \(\sqrt{3x}=\sqrt{27}\) is not an integer satisfied and \(\sqrt{x}=3\) IS an integer, BUT if \(x=2\), condition \(\sqrt{3x}=\sqrt{6}\) is not an integer satisfied and \(\sqrt{x}=\sqrt{2}\) IS NOT an integer. Two different answers. Not sufficient. Answer: A. jeeteshsingh, you should have spotted that there was something wrong with your solution as in DS two statements cannot give you TWO DIFFERENT answers to the question (as you've got). Hope it helps.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Senior Manager
Joined: 22 Dec 2009
Posts: 350

Re: Is OG Quant question answer wrong? [#permalink]
Show Tags
20 Feb 2010, 16:13
Bunuel wrote: jeeteshsingh, you should have spotted that there was something wrong with your solution as in DS two statements can not give you TWO DIFFERENT answers to the question (as you've got).
Hope it helps. My Bad.... overlooked it... Infact today I was telling this to someone over the forum that both the statements in DS would always be in sync.. Thanks Bunuel... for pointing this out.
_________________
Cheers! JT........... If u like my post..... payback in Kudos!!
Do not post questions with OAPlease underline your SC questions while postingTry posting the explanation along with your answer choice For CR refer Powerscore CR BibleFor SC refer Manhattan SC Guide
~~Better Burn Out... Than Fade Away~~



Manager
Joined: 17 Jan 2010
Posts: 143
Concentration: General Management, Strategy
GPA: 3.78
WE: Engineering (Manufacturing)

Re: Is OG Quant question answer wrong? [#permalink]
Show Tags
20 Feb 2010, 16:42
When I assumed the case \sqrt{x}=2.5 I did not take into the account that x will not be an integer in this case(x=6.25). Thanks Bunuel



Manager
Joined: 27 Oct 2009
Posts: 136
Location: Montreal
Schools: Harvard, Yale, HEC

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
29 Sep 2010, 07:09
1
This post received KUDOS
2
This post was BOOKMARKED
If x is a positive integer, is \(\sqrt{x}\) an integer?
(1) \(\sqrt{4x}\) is an integer.
(2) \(\sqrt{3x}\) is not an integer.
Last edited by ezinis on 29 Sep 2010, 09:39, edited 1 time in total.



Math Expert
Joined: 02 Sep 2009
Posts: 44388

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
29 Sep 2010, 07:29
1
This post received KUDOS
Expert's post
1
This post was BOOKMARKED
ezinis wrote: If x is a positive integer, is \sqrt{x} an integer? (1) \(\sqrt{4x}\) is an integer4. (2) \(\sqrt{3x}\) is an integer.
I am not satisfied with the official explanation. Please give yours, thanks. I think (2) should be \(\sqrt{3x}\) is NOT an integer. If \(x=integer\), is \(\sqrt{x}=integer\)? (1) \(\sqrt{4x}\) is an integer > \(2\sqrt{x}=integer\) > \(2\sqrt{x}\) to be an integer \(\sqrt{x}\) must be an integer or integer/2, but as \(x\) is an integer, then \(\sqrt{x}\) can not be integer/2, hence \(\sqrt{x}\) is an integer. Sufficient. (2)\(\sqrt{3x}\) is not an integer > if \(x=9\), condition \(\sqrt{3x}=\sqrt{27}\) is not an integer satisfied and \(\sqrt{x}=3\) IS an integer, BUT if \(x=2\), condition \(\sqrt{3x}=\sqrt{6}\) is not an integer satisfied and \(\sqrt{x}=\sqrt{2}\) IS NOT an integer. Two different answers. Not sufficient. Answer: A.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Retired Moderator
Joined: 02 Sep 2010
Posts: 779
Location: London

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
29 Sep 2010, 08:10
ezinis wrote: If x is a positive integer, is \sqrt{x} an integer? (1) \sqrt{4x} is an integer4. (2) \sqrt{3x} is an integer.
I am not satisfied with the official explanation. Please give yours, thanks. (1) \(\sqrt{4x} = 2 * \sqrt{x}\) If this is an integer, then \(\sqrt{x}\) has to be an integer (2) \(\sqrt{3x} = \sqrt{3} * \sqrt{x}\) For this to be an integer, \(\sqrt{x}\) must be of the form \(\sqrt{3} * Integer\) So \(\sqrt{x}\) is not an integer I am not sure if the question is correct as (1) and (2) are contradicting. Is it supposed to say \(\sqrt{3x}\) is not an integer ?
_________________
Math writeups 1) Algebra101 2) Sequences 3) Set combinatorics 4) 3D geometry
My GMAT story
GMAT Club Premium Membership  big benefits and savings



Math Forum Moderator
Joined: 20 Dec 2010
Posts: 1935

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
09 Jan 2011, 09:40
I didn't get the explanation: if root(4x) is an integer fact. Then 2 * root(x) is an integer. So what!!!!!!!!!! This doesn't mean that root(x) should be an integer. Because root(x) can be 1.5 and yet won't distort the fact that 2 * 1.5 = 3 is an integer. So, if x = 2.25(a non integer) root(x)=1.5 and 2*1.5=3 is an integer. if x=4(an integer) root(x)=2 and 2*2=4 is also an integer. So, statement one would be true for two values of x (2.25 and 4). root(2.25) is 1.5, not an integer. root(4) is 2, an integer. This statement is insufficient to conclude whether root(x) is an integer. What's wrong with my explanation??
_________________
~fluke
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 44388

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
09 Jan 2011, 09:52
6
This post received KUDOS
Expert's post
8
This post was BOOKMARKED
fluke wrote: I didn't get the explanation:
if root(4x) is an integer fact. Then 2 * root(x) is an integer. So what!!!!!!!!!! This doesn't mean that root(x) should be an integer. Because root(x) can be 1.5 and yet won't distort the fact that 2 * 1.5 = 3 is an integer.
So, if x = 2.25(a non integer) root(x)=1.5 and 2*1.5=3 is an integer. if x=4(an integer) root(x)=2 and 2*2=4 is also an integer. So, statement one would be true for two values of x (2.25 and 4). root(2.25) is 1.5, not an integer. root(4) is 2, an integer. This statement is insufficient to conclude whether root(x) is an integer.
What's wrong with my explanation?? You forgot that x is a positive integer, so \(\sqrt{x}\) cannot equal to \(\frac{integer}{2}\). Generally \(\sqrt{integer}\) is either an integer or an irrational number. Complete solution: If x is a positive integer, is sqrt(x) an integerIf \(x=integer\), is \(\sqrt{x}=integer\)? (1) \(\sqrt{4x}\) is an integer > \(2\sqrt{x}=integer\) > \(2\sqrt{x}\) to be an integer \(\sqrt{x}\) must be an integer or integer/2, but as \(x\) is an integer, then \(\sqrt{x}\) can not be integer/2, hence \(\sqrt{x}\) is an integer. Sufficient. (2)\(\sqrt{3x}\) is not an integer > if \(x=9\), condition \(\sqrt{3x}=\sqrt{27}\) is not an integer satisfied and \(\sqrt{x}=3\) IS an integer, BUT if \(x=2\), condition \(\sqrt{3x}=\sqrt{6}\) is not an integer satisfied and \(\sqrt{x}=\sqrt{2}\) IS NOT an integer. Two different answers. Not sufficient. Answer: A. Hope it's clear.
_________________
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? Extrahard Quant Tests with Brilliant Analytics



Math Forum Moderator
Joined: 20 Dec 2010
Posts: 1935

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
10 Jan 2011, 01:32
good explanation Bunuel and you are right in saying that I carelessly overlooked the fact that x was a positive integer... thanks ~fluke
_________________
~fluke
GMAT Club Premium Membership  big benefits and savings



Math Expert
Joined: 02 Sep 2009
Posts: 44388

Re: Integers [#permalink]
Show Tags
28 Feb 2011, 12:56



Manager
Joined: 20 Jan 2011
Posts: 79

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
19 Jul 2011, 06:29
Quote: then sqrt(x) can not be integer/2 I think sqrt(x) can be integer/2 as long as (integer/2) itself is an integer i.e. that integer is multiple of 2. I think that is what bunuel meant. And the answer remains same.
_________________
Conquer the Hell and make it Haven. Brain is your hell and Success is your haven!
"Kudos" is significant part of GMAT prep. If you like it, you just click it



Manager
Joined: 12 Oct 2011
Posts: 241

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
03 Jan 2012, 13:06
Oh nice problem. Took quite some time to answer it but got A. Explanation by Bunuel is more than sufficient to understand the solution.
_________________
Consider KUDOS if you feel the effort's worth it



Manager
Joined: 29 Jul 2011
Posts: 103
Location: United States

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
03 Jan 2012, 14:31
Both equations and number plugging helps here. 1. sqrt(4x) = integer, that is 2 x sqrt(x) = integer. For this equation to be true, sqrt(x) has to be an integer. SUFFICIENT. If number plugging, use 4x4 and 4x9 combinations 2. sqrt(3x) = frac. Here, sqrt(3) x sqrt(x) = frac, that is frac x sqrt(x) = frac. Difficult to determine if this relationship can infer sqrt(x) as integer. So, let's go with number plugging. sqrt(3x4) = 12 satisfies, and sqrt(4) = integer. sqrt(3x5) = 15 satisfies, but sqrt(5) = frac. So, insufficient. +1 for A.
_________________
I am the master of my fate. I am the captain of my soul. Please consider giving +1 Kudos if deserved!
DS  If negative answer only, still sufficient. No need to find exact solution. PS  Always look at the answers first CR  Read the question stem first, hunt for conclusion SC  Meaning first, Grammar second RC  Mentally connect paragraphs as you proceed. Short = 2min, Long = 34 min



Manager
Joined: 21 Jul 2012
Posts: 68

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
15 Dec 2012, 11:21
Bunuel  can you further explain your explanation for Statement 1? I am confused because if x is a positive integer, sqrt(x) can equal an integer/2 if for example x was equal to 4. 4 is a positive integer and the square root of 4 is equal to 4/2. I think I am missing the overall takeaway, can you help clarify? Thanks!



Math Expert
Joined: 02 Sep 2009
Posts: 44388

Re: If x is a positive integer, is x^1/2 an integer [#permalink]
Show Tags
16 Dec 2012, 08:37




Re: If x is a positive integer, is x^1/2 an integer
[#permalink]
16 Dec 2012, 08:37



Go to page
1 2
Next
[ 38 posts ]



