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

Question

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.

Bunuel · Accepted Answer

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.

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mau5 · Answer

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}}   	o t = z^{\frac{3+1}{3}} 	o 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.

shekharverma · Answer

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

Bunuel · Answer

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.

NoHalfMeasures · Answer

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?

vad3tha · Answer

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.

taransambi · Answer

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.

honchos · Answer

Ratio of two Integers is never an Irrational number, right?

Bunuel · Answer

Yes.  In mathematics, an irrational number is any real number that cannot be expressed as a ratio of integers. Irrational numbers cannot be represented as terminating or repeating decimals.

DanielAustin · Answer

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

THUS answer E.

How is my thinking flawed here?

Bunuel · Answer

If  x=2^{24}  it's still is the 12th power of an integer:  x=2^{24}=4^{12}

JeffTargetTestPrep · Answer

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.

Answer: C

GMATGuruNY · Answer

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.

C

Alternate approach:

Statement 1: x = a³, where a is an integer 
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.

C

nutella · Answer

Bunuel 
can you help me understand this:
"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"
Thanks

Bunuel · Answer

Consider the following examples: 2^3, 6^6 = (6^2)^3 = 2^6*3^6, 10^12 = (10^4)^3 = 2^12*5^12, ... All those numbers are perfect squares and the powers of the primes of all those numbers are multiples of 3.

Tonynguyen152 · Answer

I have an example here:
(1) x=m^5 -> x^7=m^35
(2) x=n^7 -> x^5=n^35

(2)/(1) -> x^2=(n/m)^35
x is not yet proven to be equal to the 35th power of an integer

Could you explain more about this take away?

MBAHOUSE · Answer

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

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

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

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