Data Sufficiency

This question came from the new GMAT Prep software:

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

OA:
C

I get the explanation for the correct answer but at the same time don't quite understand what x can be. From 1) we get x=m^3 and from 2) x=n^4. Taken together m^3=n^4=x. Given the ask of, I suppose knowing x is not important? But what can x be to satisfy such conditions?

Many thanks for your thoughts/help! [/spoiler]

lamania wrote:This question came from the new GMAT Prep software:

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

OA:
C

I get the explanation for the correct answer but at the same time don't quite understand what x can be. From 1) we get x=m^3 and from 2) x=n^4. Taken together m^3=n^4=x. Given the ask of, I suppose knowing x is not important? But what can x be to satisfy such conditions?

Many thanks for your thoughts/help! [/spoiler]

Hi lamania,

The answer is [C]. Here, is the explanation-

Its quite obvious that neither of them alone can be sufficient. So, we check for [C] (both together sufficient) or [E] both together not sufficient.

Proceeding with [C]..

x = a^3
taking 4th power of both sides..
x^4 = a^12 ... (1.)

x= b^4
taking 3rd power of both sides..
x^3 = b^12 ... (2.)

Now, dividing (1.) by (2.) we have-
x = (a/b)^12 ..

Here, a/b has to be an integer hence [C] is the correct choice.

Hope it helps!!

Regards,
Shantanu

Dividing the two! I shall remember that tactic! thanks so much.

May I ask why a/b must be an integer? x^4/x^3=(a/b)^12 was sufficient to answer the question. How can one be sure a/b is an integer?

thanks again

lamania wrote:This question came from the new GMAT Prep software:

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

OA:
C

I get the explanation for the correct answer but at the same time don't quite understand what x can be. From 1) we get x=m^3 and from 2) x=n^4. Taken together m^3=n^4=x. Given the ask of, I suppose knowing x is not important? But what can x be to satisfy such conditions?

Many thanks for your thoughts/help! [/spoiler]

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

x can take any value. INSUFF

2) x is equal to the 4th power of an integer

x can take any value. INSUFF

Combining both a^3 = a^4 = x. Remember that 1^n = 1. No other number can satisfy this criteria. Once you read that a number raised to two different powers give same value then it has to be one. When you are given two even powers then you should be careful as it can be -1 as well Hope this helps.

IMO C.

Thanks I thought of the same thing too but the question stem says x is greater than 1 so 1 was not workable. The division approach shantanu86 mentioned was a good approach and we need not identify what x is.

lamania wrote:Dividing the two! I shall remember that tactic! thanks so much.

May I ask why a/b must be an integer? x^4/x^3=(a/b)^12 was sufficient to answer the question. How can one be sure a/b is an integer?

thanks again

Hi lamania,

Here, we *need* to establish that a/b is an integer because question specifically asks- "is x equal to the 12th power of an integer?".

Also, in general, every positive number x will always have a 12th root, so if integer was not mentioned we wouldn't have required any information

Intuitively its much easier to visualize that a/b has to be an integer.
But it can be mathematically proved too-

b^4 = (a/b)^12 => (a/b)^3 = b, an integer.. (1.)

Since a and b are integers and a/b has to be a rational number.
Also, third power of a rational number is integer only when the number itself is rational.. it implies that (a/b) is an integer.

Let me know if you have doubts in this explanation.

Regards,
Shantanu

sam2304 wrote:

lamania wrote:This question came from the new GMAT Prep software:

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

OA:
C

I get the explanation for the correct answer but at the same time don't quite understand what x can be. From 1) we get x=m^3 and from 2) x=n^4. Taken together m^3=n^4=x. Given the ask of, I suppose knowing x is not important? But what can x be to satisfy such conditions?

Many thanks for your thoughts/help! [/spoiler]

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

x can take any value. INSUFF

2) x is equal to the 4th power of an integer

x can take any value. INSUFF

Combining both a^3 = a^4 = x. Remember that 1^n = 1. No other number can satisfy this criteria. Once you read that a number raised to two different powers give same value then it has to be one. When you are given two even powers then you should be careful as it can be -1 as well Hope this helps.

IMO C.

Hi sam2304,

The problem in the approach you suggested is that the question no where says that integer in info 1 and 2 is same.
If it were same x=1 is the only workable solution.. (not allowed as x>1)

Regards,
Shantanu

shantanu86 wrote:
Hi sam2304,

The problem in the approach you suggested is that the question no where says that integer in info 1 and 2 is same.
If it were same x=1 is the only workable solution.. (not allowed as x>1)

Regards,
Shantanu

Yup, I get it buddy. Thanks for pointing it out. I missed that information, my bad

lamania wrote:This question came from the new GMAT Prep software:

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

Common sense and logic are some of the most underrated GMAT tools; applying "math common sense" to this problem makes it understandable.

Let's agree that neither statement on its own is sufficient and jump right to combination.

Since x > 1, we know that statements (1) and (2) have to be referring to different integers (because there's no integers other than 0 and 1 for which x^3 = x^4).

So, how could one number be both the cube AND the fourth power of an integer? Only if the number of identical factors of x is a multiple of both 3 and 4.

In other words, if we call each identical factor a, we get:

x = (a*a*a*a)(a*a*a*a)(a*a*a*a) = (a^4)^3 (so x is a perfect cube)

and

x = (a*a*a)(a*a*a)(a*a*a)(a*a*a) = (a^3)^4 (so x is a perfect power of 4)

Since x must have 12 factors of a for this to hold true, x MUST be a perfect power of 12.

Choose (C)!

* * *

As an aside, there are an infinite number of possible values of x, since a could be any integer.

The statements still do not say that the integers referred to here are same. Hence I think (E) is answer.

ronnie1985 wrote:The statements still do not say that the integers referred to here are same. Hence I think (E) is answer.

You are getting confused mate, see stuart's explanation very carefully. They are all different integers.

1) x is equal to the 3rd power of an integer
x = (a*a*a*a)(a*a*a*a)(a*a*a*a) = (a^4)^3
Here a*a*a*a - is one integer
x = 4096 = 16^3

2) x is equal to the 4th power of an integer
x = (a*a*a)*(a*a*a)*(a*a*a)*(a*a*a) = (a^4)^3
Here a*a*a - is the integer
x = 4096 = 8^4

x = 4096 = 2^12

Stuart Kovinsky wrote:

lamania wrote:This question came from the new GMAT Prep software:

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

Common sense and logic are some of the most underrated GMAT tools; applying "math common sense" to this problem makes it understandable.

Let's agree that neither statement on its own is sufficient and jump right to combination.

Since x > 1, we know that statements (1) and (2) have to be referring to different integers (because there's no integers other than 0 and 1 for which x^3 = x^4).

So, how could one number be both the cube AND the fourth power of an integer? Only if the number of identical factors of x is a multiple of both 3 and 4.

In other words, if we call each identical factor a, we get:

x = (a*a*a*a)(a*a*a*a)(a*a*a*a) = (a^4)^3 (so x is a perfect cube)

and

x = (a*a*a)(a*a*a)(a*a*a)(a*a*a) = (a^3)^4 (so x is a perfect power of 4)

Since x must have 12 factors of a for this to hold true, x MUST be a perfect power of 12.

Choose (C)!

* * *

As an aside, there are an infinite number of possible values of x, since a could be any integer.

i'm sorry stuart, i'm not sure i understand, how does a^3 = a^4 in your explanation?

fangtray wrote: i'm sorry stuart, i'm not sure i understand, how does a^3 = a^4 in your explanation?

See the post above yours. I posted an example which helps to understand stuart's explanation.

Let me take a try :

* I think it is pretty clear that (A), (B) and (D) are out of the picture
** I cannot use mathematical expressions, so here goes a simple reading tip for my answer below - CR(k) means Cube Root of K. Okay ? Now read on ...

Stem States : X is an integer greater than 1 -> X belongs to the positive integer set & X>1

1) States X = a^3 -> a is said to be an integer
2) States X = b^4 -> b is said to be an integer

Using both (1) and (2), It is safe to say a^3 = b^4, right ?
That means a = b*CR(b), right ?
But a is an integer and b is an integer , so CR(b) must be an integer. There is a theorem for this on squares and cubes. But the gist is that this has to hold true for cubes and 4th powers, i reckon.

now the problem is solved to taking the 12th power of CR(b).
(CR(b))^12 = (b)^4 -> Since (CR(b))^3 = b
(b)^4 = X -> Given by Statement 2
Hence yes -> C.

Example : X = 16*16*16 & X = 8*8*8*8

HTH!

Stuart's explaination is easy to understand.

its nothing but x = (a^3)^4 = (a^4)^3 = (a^12) (Simply rules of power)

other example: it can also be x = (a^2)^3 = (a^3)^2 = (a^6)