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

It is currently 27 Jun 2019, 03:31

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

If x is an integer greater than 1, is x equal to the 12th

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

Hide Tags

 
Intern
Intern
avatar
Joined: 14 Nov 2010
Posts: 3
If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 23 Jun 2012, 10:44
12
60
00:00
A
B
C
D
E

Difficulty:

  45% (medium)

Question Stats:

61% (01:30) correct 39% (01:38) wrong based on 1036 sessions

HideShow timer Statistics


If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer

(2) x is equal to the 4th Power of an integer.
Most Helpful Expert Reply
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 23 Jun 2012, 11:15
60
40
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> \(x=m^3\) for some positive integer \(m\). If \(m\) itself is 4th power of some integer (for example if \(m=2^4\)), then the answer will be YES (since in this case \(x=(2^4)^3=2^{12}\)), but if it's not (for example if \(m=2\)), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that \(x^4=m^{12}\).

(2) x is equal to the 4th Power of an integer --> \(x=n^4\) for some positive integer \(n\). If \(n\) itself is 3rd power of some integer (for example if \(n=2^3\)), then the answer will be YES (since in this case \(x=(2^3)^4=2^{12}\)), but if it's not (for example if \(n=2\)), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that \(x^3=n^{12}\).

(1)+(2) Divide (i) by (ii): \(x=(\frac{m}{n})^{12}=integer\). Now, \(\frac{m}{n}\) can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore \(\frac{m}{n}\) must be an integer, hence \(x=(\frac{m}{n})^{12}=integer^{12}\). Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html
_________________
Most Helpful Community Reply
Verbal Forum Moderator
User avatar
B
Joined: 10 Oct 2012
Posts: 606
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 31 Jan 2013, 00:13
10
4
Given that x>1 and an integer.

From F.S 1, we have \(x=t^3\),t is a positive integer. Now for t=16, we will have a sufficient condition but not for say t=8. Thus not sufficient.

From F.S 2, we have \(x=z^4\). z is a positive integer. Now just as above, for z=8, we will have a sufficient condition but not for say z=16. Thus not sufficient.

Combining both of them, we have;

\(x=t^3; x=z^4\). Hence, \(t^3 = z^4\). Now this can be written as \(t = z^{\frac{4}{3}}\) \(\to t = z^{\frac{3+1}{3}} \to t = z*z^{\frac{1}{3}}\)
Now, as both t and z are integers, we must have \(z^{\frac{1}{3}}\) as an integer.Thus, t = kz , where \(k = z^{\frac{1}{3}}\)
Cubing on both sides, we have
\(z = k^3.\)

Replace this value of z,\(x = z^4 or x = (k^3)^4 = k^{12}\).

C.
_________________
General Discussion
Intern
Intern
avatar
Joined: 04 Jun 2012
Posts: 3
Concentration: General Management, Finance
GMAT Date: 07-23-2012
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 03 Jul 2012, 08:47
1
2
By Statement I: x= m^3 not sufficient since m= 16 satisfies but m= 8 does not satisfy
By Statement II: x= n^4 not sufficient since n= 8 satisfies but n = 16 does not satisfy
By I & II: X= m^3 and x = n^4 => m^3= n^4 which is only true for 1 or 0, in both cases original condition is satisfied
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 03 Jul 2012, 09:01
1
shekharverma wrote:
By Statement I: x= m^3 not sufficient since m= 16 satisfies but m= 8 does not satisfy
By Statement II: x= n^4 not sufficient since n= 8 satisfies but n = 16 does not satisfy
By I & II: X= m^3 and x = n^4 => m^3= n^4 which is only true for 1 or 0, in both cases original condition is satisfied


Notice that we are told that \(x\) is an integer greater than 1, so \(m=n=0\) or \(m=n=1\) are not possible since in this case \(x\) becomes 0 or 1.

Though if we proceed the way you propose, then from \(x=m^3\) and \(x=n^4\) we can conclude that those two conditions also hold true when \(m=a^{4}\) and \(n=a^3\) (for some positive integer \(a\)), so when \(x=m^3=n^4=a^{12}\).

Hope it helps.
_________________
Retired Moderator
avatar
Joined: 05 Sep 2010
Posts: 647
GMAT ToolKit User
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 15 Aug 2012, 09:39
i agree that indivisualy we cannot answer this question ...but how abt this approach .if we combine both statement then we can be sure that x =(int ) ^12 becoz under this condition only can both the conditions be met .so we can now be sure that this int can be expressed as some int raised to the power of 12 .expert plz evaluate this !!
Director
Director
avatar
Joined: 29 Nov 2012
Posts: 732
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 27 Jan 2013, 21:42
Bunuel wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> \(x=m^3\) for some positive integer \(m\). If \(m\) itself is 4th power of some integer (for example if \(m=2^4\)), then the answer will be YES (since in this case \(x=(2^4)^3=2^{12}\)), but if it's not (for example if \(m=2\)), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that \(x^4=m^{12}\).

(2) x is equal to the 4th Power of an integer --> \(x=n^4\) for some positive integer \(n\). If \(n\) itself is 3rd power of some integer (for example if \(n=2^3\)), then the answer will be YES (since in this case \(x=(2^3)^4=2^{12}\)), but if it's not (for example if \(n=2\)), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that \(x^3=n^{12}\).

(1)+(2) Divide (i) by (ii): \(x=(\frac{m}{n})^{12}=integer\). Now, \(\frac{m}{n}\) can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore \(\frac{m}{n}\) must be an integer, hence \(x=(\frac{m}{n})^{12}=integer^{12}\). Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


Is this true in all cases that it must be an integer ( is there a theorem or something along those lines) , could you please provide an example.
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 28 Jan 2013, 00:56
1
fozzzy wrote:
Bunuel wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> \(x=m^3\) for some positive integer \(m\). If \(m\) itself is 4th power of some integer (for example if \(m=2^4\)), then the answer will be YES (since in this case \(x=(2^4)^3=2^{12}\)), but if it's not (for example if \(m=2\)), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that \(x^4=m^{12}\).

(2) x is equal to the 4th Power of an integer --> \(x=n^4\) for some positive integer \(n\). If \(n\) itself is 3rd power of some integer (for example if \(n=2^3\)), then the answer will be YES (since in this case \(x=(2^3)^4=2^{12}\)), but if it's not (for example if \(n=2\)), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that \(x^3=n^{12}\).

(1)+(2) Divide (i) by (ii): \(x=(\frac{m}{n})^{12}=integer\). Now, \(\frac{m}{n}\) can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore \(\frac{m}{n}\) must be an integer, hence \(x=(\frac{m}{n})^{12}=integer^{12}\). Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


Is this true in all cases that it must be an integer ( is there a theorem or something along those lines) , could you please provide an example.


What you mean by "all cases"? Anyway, if m and n are integers and \(x=(\frac{m}{n})^{12}=integer\), then m/n=integer.
_________________
Retired Moderator
avatar
Joined: 29 Oct 2013
Posts: 257
Concentration: Finance
GPA: 3.7
WE: Corporate Finance (Retail Banking)
GMAT ToolKit User
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 21 Nov 2013, 21:12
2
2
So the important take away here is: if X = nth power of an integer and x= mth power of an integer simultaneously, x= (LCM of m and n)th power of an integer?
_________________
Please contact me for super inexpensive quality private tutoring

My journey V46 and 750 -> http://gmatclub.com/forum/my-journey-to-46-on-verbal-750overall-171722.html#p1367876
SVP
SVP
User avatar
Joined: 06 Sep 2013
Posts: 1651
Concentration: Finance
GMAT ToolKit User
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 27 Dec 2013, 09:52
1
MensaNumber wrote:
So the important take away here is: if X = nth power of an integer and x= mth power of an integer simultaneously, x= (LCM of m and n)th power of an integer?


Well that's what I'm asking myself but think about it for a sec

For perfect cube we need all prime factors to have a multiple of 3
For perfect fourth powers we need all the same prime factors to have a multiple of 4

Hence, for both we need all the prime factors to have multiples of 12 at least

So IMHO I think this should be correct under this scenario

Bunuel, would you give your blessing on this statement?

Cheers!
J :)
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 13 May 2014, 00:56
jlgdr wrote:
MensaNumber wrote:
So the important take away here is: if X = nth power of an integer and x= mth power of an integer simultaneously, x= (LCM of m and n)th power of an integer?


Well that's what I'm asking myself but think about it for a sec

For perfect cube we need all prime factors to have a multiple of 3
For perfect fourth powers we need all the same prime factors to have a multiple of 4

Hence, for both we need all the prime factors to have multiples of 12 at least

So IMHO I think this should be correct under this scenario

Bunuel, would you give your blessing on this statement?

Cheers!
J :)


Yes, that's correct.
_________________
Manager
Manager
User avatar
Joined: 22 Feb 2009
Posts: 159
GMAT ToolKit User
Re: Data Sufficiency problem - exponents  [#permalink]

Show Tags

New post 07 Aug 2014, 21:26
7
1
taransambi wrote:
Source: Question Pack 1

If X is an integer greater than 1, is X equal to 12th power of an integer?

1. X is equal to 3rd power of an integer.
2. X is equal to 4th power for an integer.


Statement 1: x= a^3. For example, x = 2^3 = 8 --> cannot equal to 12th power of an integer--> INSUFFICIENT
Statement 2: x= a^4. For example, x = 2^4 = 16--> cannot equal to 12th power of an integer--> INSUFFICIENT
Combine 2 statements:
x= a^3 --> x^4= a^12
x= b^4 --> x^3=b^12
-> x^4/x^3 = x = a^12/b^12 = (a/b)^12
x is an integer, so (a/b)^12 is an integer, so (a/b) has to be an integer also, called c
so x= c^12 --> SUFFICIENT

C is the answer.
Hope it helps.
_________________
.........................................................................
+1 Kudos please, if you like my post
Intern
Intern
avatar
Joined: 22 Mar 2013
Posts: 18
Concentration: Operations, Entrepreneurship
GMAT 1: 620 Q47 V28
GMAT 2: 680 Q45 V38
WE: Engineering (Manufacturing)
Reviews Badge
Re: Data Sufficiency problem - exponents  [#permalink]

Show Tags

New post 07 Aug 2014, 22:11
3
If X is an integer greater than 1, is X equal to 12th power of an integer?

1. X is equal to 3rd power of an integer.
2. X is equal to 4th power for an integer.


Here is how i solved it. From statements 1 and 2, we know that X=a^3 as well as b^4. Therefore, a^3=b^4.

This is only possible when either 1) a=b=1 OR 2) a=b=0.

The questions says that X>1, so none of the above cases are true.

So, for a^3 to be equal to b^4, a needs to have a 4th power of b in it AND b needs to have a 3rd power of a in it. In either case, X will have a 12th power of an integer in it. Hence, C.
Senior Manager
Senior Manager
User avatar
Status: Verbal Forum Moderator
Joined: 17 Apr 2013
Posts: 463
Location: India
GMAT 1: 710 Q50 V36
GMAT 2: 750 Q51 V41
GMAT 3: 790 Q51 V49
GPA: 3.3
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 04 Jun 2015, 09:16
Bunuel wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> \(x=m^3\) for some positive integer \(m\). If \(m\) itself is 4th power of some integer (for example if \(m=2^4\)), then the answer will be YES (since in this case \(x=(2^4)^3=2^{12}\)), but if it's not (for example if \(m=2\)), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that \(x^4=m^{12}\).

(2) x is equal to the 4th Power of an integer --> \(x=n^4\) for some positive integer \(n\). If \(n\) itself is 3rd power of some integer (for example if \(n=2^3\)), then the answer will be YES (since in this case \(x=(2^3)^4=2^{12}\)), but if it's not (for example if \(n=2\)), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that \(x^3=n^{12}\).

(1)+(2) Divide (i) by (ii): \(x=(\frac{m}{n})^{12}=integer\). Now, \(\frac{m}{n}\) can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore \(\frac{m}{n}\) must be an integer, hence \(x=(\frac{m}{n})^{12}=integer^{12}\). Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


What if m/n = √2 type that is quite possible.
_________________
Like my post Send me a Kudos :) It is a Good manner.
My Debrief: http://gmatclub.com/forum/how-to-score-750-and-750-i-moved-from-710-to-189016.html
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 04 Jun 2015, 09:21
honchos wrote:
Bunuel wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> \(x=m^3\) for some positive integer \(m\). If \(m\) itself is 4th power of some integer (for example if \(m=2^4\)), then the answer will be YES (since in this case \(x=(2^4)^3=2^{12}\)), but if it's not (for example if \(m=2\)), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that \(x^4=m^{12}\).

(2) x is equal to the 4th Power of an integer --> \(x=n^4\) for some positive integer \(n\). If \(n\) itself is 3rd power of some integer (for example if \(n=2^3\)), then the answer will be YES (since in this case \(x=(2^3)^4=2^{12}\)), but if it's not (for example if \(n=2\)), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that \(x^3=n^{12}\).

(1)+(2) Divide (i) by (ii): \(x=(\frac{m}{n})^{12}=integer\). Now, \(\frac{m}{n}\)can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore \(\frac{m}{n}\) must be an integer, hence \(x=(\frac{m}{n})^{12}=integer^{12}\). Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


What if m/n = √2 type that is quite possible.


Have you read the highlighted part?
_________________
Senior Manager
Senior Manager
User avatar
Status: Verbal Forum Moderator
Joined: 17 Apr 2013
Posts: 463
Location: India
GMAT 1: 710 Q50 V36
GMAT 2: 750 Q51 V41
GMAT 3: 790 Q51 V49
GPA: 3.3
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 04 Jun 2015, 09:26
Bunuel wrote:
honchos wrote:
Bunuel wrote:
If x is an integer greater than 1, is x equal to the 12th power of an integer ?

(1) x is equal to the 3rd Power of an integer --> \(x=m^3\) for some positive integer \(m\). If \(m\) itself is 4th power of some integer (for example if \(m=2^4\)), then the answer will be YES (since in this case \(x=(2^4)^3=2^{12}\)), but if it's not (for example if \(m=2\)), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that \(x^4=m^{12}\).

(2) x is equal to the 4th Power of an integer --> \(x=n^4\) for some positive integer \(n\). If \(n\) itself is 3rd power of some integer (for example if \(n=2^3\)), then the answer will be YES (since in this case \(x=(2^3)^4=2^{12}\)), but if it's not (for example if \(n=2\)), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that \(x^3=n^{12}\).

(1)+(2) Divide (i) by (ii): \(x=(\frac{m}{n})^{12}=integer\). Now, \(\frac{m}{n}\)can be neither an irrational number (since it's the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore \(\frac{m}{n}\) must be an integer, hence \(x=(\frac{m}{n})^{12}=integer^{12}\). Sufficient.

Answer: C.

Hope it's clear.

P.S. Please read and follow: rules-for-posting-please-read-this-before-posting-133935.html


What if m/n = √2 type that is quite possible.


Have you read the highlighted part?


Ratio of two Integers is never an Irrational number, right?
_________________
Like my post Send me a Kudos :) It is a Good manner.
My Debrief: http://gmatclub.com/forum/how-to-score-750-and-750-i-moved-from-710-to-189016.html
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 04 Jun 2015, 09:29
Current Student
User avatar
B
Status: DONE!
Joined: 05 Sep 2016
Posts: 368
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 23 Sep 2016, 09:53
C is correct. Here's why:

(1) x = n^3 --> Try plugging in arbitrary value for n.

x = 2^3 --> x = 8 --> now ask yourself what integer raised to the 12th power could give you 8. Answer = none

INSUFFICIENT

(2) x = n^4 --> repeat the same process as in (1)

x = 2^4 --> x= 16 --> There is no integer that could give us this value using main equation

INSUFFICIENT

(1) + (2) Together - SUFFICIENT
Manager
Manager
User avatar
B
Joined: 28 Sep 2013
Posts: 81
GMAT 1: 740 Q51 V39
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 08 Dec 2016, 15:23
1
Bunuel wrote:
[b] Now, \(\frac{m}{n}\) can be neither an irrational number (since it's the ratio of two integers)


I am unable to understand this part can you please explain or perhaps provide me a link where I can understand my knowledge gap. Thanks!
_________________
Richa Champion | My GMAT Journey - 470 720 740

Target 760+

Not Improving after Multiple attempts. I can guide You.
Contact me richacrunch2@gmail.com
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 55804
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

Show Tags

New post 09 Dec 2016, 02:55
RichaChampion wrote:
Bunuel wrote:
[b] Now, \(\frac{m}{n}\) can be neither an irrational number (since it's the ratio of two integers)


I am unable to understand this part can you please explain or perhaps provide me a link where I can understand my knowledge gap. Thanks!


Check here: if-n-p-q-p-and-q-are-nonzero-integers-is-an-integer-101475.html and here: is-a-even-133175.html
_________________
GMAT Club Bot
Re: If x is an integer greater than 1, is x equal to the 12th   [#permalink] 09 Dec 2016, 02:55

Go to page    1   2    Next  [ 27 posts ] 

Display posts from previous: Sort by

If x is an integer greater than 1, is x equal to the 12th

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


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