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

 It is currently 22 Oct 2018, 13:52

### 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

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

Author Message
TAGS:

### Hide Tags

Manager
Joined: 23 Jan 2016
Posts: 199
Location: India
GPA: 3.2
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

13 Dec 2016, 11:28
can the answer not be as simple as - to be the 12th power of an integer, x would need to be the 3rd power and 4th power as well. Since individually the statements do not tell us this information, if we combine we get that x= m^12. Experts could you please let me know if this approach is wrong.

Thanks.
Senior Manager
Status: Come! Fall in Love with Learning!
Joined: 05 Jan 2017
Posts: 472
Location: India
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

23 Feb 2017, 06:01
Prompt Analysis
x is an integer.

Superset
The answer to this question will be either yes or no.

Translation
In order to know if x = p^12, we need:
1# exact value of p or x
2# any property or equation to infer the statement.

Statement analysis

St 1: x =a^3. Insufficient as a may or may not be the form of b^4. Hence option a, d eliminated.
St 2: x = c^4. Insufficient as a may or may not be the form of d^4. Hence b is eliminated.

St 1 & St 2: LCM of 4 and 3 is 12. Hence x is an integer to the power 12. Option C.
_________________

GMAT Mentors

Intern
Joined: 24 Oct 2017
Posts: 2
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

08 Nov 2017, 09:32
Hi, I need some help.

I follow the above posts on (i) and (ii) as not sufficient. However, I am struggling with a case where, if true, leads to the answer E, not C:

If $$x=m^3=n^4$$, where $$m=2^8$$ and $$n=2^6$$, then

$$x=(2^8)^3=(2^6)^4=2^{24}$$ which does NOT equal $$2^{12}$$; NOT SUFFICIENT

But where where $$m=2^4$$ and $$n=2^3$$, then

$$x=(2^4)^3=(2^3)^4=2^{12}$$, SUFFICIENT

How is my thinking flawed here?
Math Expert
Joined: 02 Sep 2009
Posts: 50042
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

08 Nov 2017, 09:44
1
DanielAustin wrote:
Hi, I need some help.

I follow the above posts on (i) and (ii) as not sufficient. However, I am struggling with a case where, if true, leads to the answer E, not C:

If $$x=m^3=n^4$$, where $$m=2^8$$ and $$n=2^6$$, then

$$x=(2^8)^3=(2^6)^4=2^{24}$$ which does NOT equal $$2^{12}$$; NOT SUFFICIENT

But where where $$m=2^4$$ and $$n=2^3$$, then

$$x=(2^4)^3=(2^3)^4=2^{12}$$, SUFFICIENT

How is my thinking flawed here?

If $$x=2^{24}$$ it's still is the 12th power of an integer: $$x=2^{24}=4^{12}$$
_________________
Target Test Prep Representative
Affiliations: Target Test Prep
Joined: 04 Mar 2011
Posts: 2830
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

12 Nov 2017, 08:42
bhupi 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

(2) x is equal to the 4th Power of an integer.

We are given that x is an integer greater than 1 and must determine whether x is equal to the 12th power of an integer.

Statement One Alone:

x is equal to the 3rd power of an integer.

Using the information in statement one, we cannot determine whether x is equal to the 12th power of an integer. For example, if x = 8 = 2^3, then it’s not equal to the 12th power of an integer. However, if x = (2^4)^3 = 2^12, then it is equal to the 12th power of an integer. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x is equal to the 4th power of an integer

Using the information in statement two, we cannot determine whether x is equal to the 12th power of an integer. For example, if x = 16 = 2^4, then it’s not equal to the 12th power of an integer. However, if x = (2^3)^4 = 2^12, then it is equal to the 12th power of an integer. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the information from statements one and two, we know that x is equal to the 3rd power of an integer and that x is also equal to the 4th power of some other integer. Let’s represent x as a^3 where a is an integer > 1. Since a^3 is also a 4th power, the 4th root of a^3 is an integer. The only way this could happen is if a is also the 4th power of an integer; in other words, a by itself is a 4th power, say a = b^4 where b is an integer > 1.

Thus, x = a^3 = (b^4)^3 = b^12. Therefore, x is equal to the 12th power of an integer.

_________________

Jeffery Miller

GMAT Quant Self-Study Course
500+ lessons 3000+ practice problems 800+ HD solutions

Intern
Joined: 03 Aug 2017
Posts: 11
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

16 Oct 2018, 13:08
I believe that for this question one has to realize that the number is both a power of 3 and 4 and being 12 the LCM it is indeed a power of 12th..
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.

Hope it's clear.

Senior Manager
Joined: 04 Aug 2010
Posts: 296
Schools: Dartmouth College
Re: If x is an integer greater than 1, is x equal to the 12th  [#permalink]

### Show Tags

16 Oct 2018, 13:27
bhupi 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

(2) x is equal to the 4th Power of an integer.

Test an EASY CASE.
Test POWERS OF 2.

Statement 1:
x = 2³, 2⁶, 2⁹, 2¹²...
If x = 2³, then x is NOT equal to the 12th power of an integer.
If x = 2¹², then x IS equal to the 12th power of an integer.
INSUFFICIENT.

Statement 2:
x = 2⁴, 2⁸, 2¹²...
If x = 2⁴, then x is NOT equal to the 12th power of an integer.
If x = 2¹², then x IS equal to the 12th power of an integer.
INSUFFICIENT.

Statements combined:
The smallest value common to both the red list and the blue list is 2¹², which is the 12th power of an integer.
If we extend the two lists, we get:
x = 2¹⁵, 2¹⁸, 2²¹, 2²⁴...
x = 2¹⁶, 2²⁰, 2²⁴...
The next value common to both lists is 2²⁴ = 4¹², which is the 12th power of an integer.
Implication:
To satisfy both statements, x must be the 12th power of an integer.
SUFFICIENT.

Alternate approach:

Statement 1: x = a³, where a is an integer[/b]
If a=2, then x = 2³, which is not the 12th power of an integer.
If a=2⁴, then x = (2⁴)³ = 2¹², which is the 12th power of an integer.
INSUFFICIENT.

Statement 2: x = b⁴, where b is an integer
If b=2, then x = 2⁴, which is not the 12th power of an integer.
If b=2³, then x = (2³)⁴ = 2¹², which is the 12th power of an integer.
INSUFFICIENT.

Statements 1 and 2 combined:

Since x = a³ and x = b⁴, we get:
a³ = b⁴
a³ = (b³)b
b = (a/b)³.
Since b is an integer, (a/b)³ is an integer.
Since a/b = integer/integer -- the definition of a rational number -- it is not possible that a/b is equal to an irrational value such as ³√2.
Thus, in order for (a/b)³ to be an integer, a/b must be an integer, implying that b is the CUBE OF AN INTEGER.
Thus, x = b⁴ = (integer³)⁴ = integer¹².
SUFFICIENT.

_________________

GMAT and GRE Tutor
Over 1800 followers
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.

Re: If x is an integer greater than 1, is x equal to the 12th &nbs [#permalink] 16 Oct 2018, 13:27

Go to page   Previous    1   2   [ 27 posts ]

Display posts from previous: Sort by