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Source: — Data Sufficiency |
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by GMATGuruNY » Sat Jun 23, 2012 3:08 pm
Bhupisuhag 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.

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=2^4, then x = (2^4)³ = 2^12, which is the 12th power of an integer.
INSUFFICIENT.

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

Statements 1 and 2 combined:
Since x = a³ and x = b^4, we get:
a³ = b^4
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 = b^4 = (integer³)^4 = integer^12.
SUFFICIENT.

The correct answer is C.
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by sandeep_thaparianz » Sun Jun 24, 2012 4:55 am
Great explanation above OA is C
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by GMATsid2016 » Mon Dec 12, 2016 9:12 am
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 = b^4 = (integer³)^4 = integer^12.
SUFFICIENT.
Hi GMATGuruNY ,

Everything is clear. can you please explain the above part?

Thanks,

Sid
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by Jay@ManhattanReview » Thu Dec 15, 2016 1:09 am
Bhupisuhag 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.
It is clear that each statement is insufficient.

Now let's combine the two.

S1: Say, X = K^3 => X^(1/3) = K; K is an integer

S2: Say, X = M^4 => X^(1/4) = M; M is an integer

=> X^(1/3) * X^(1/4) = K*M = N; N is an integer (Product of two integers, K and M is integer)

=> X^(1/12) = N

=> X equal to the 12th power of an integer.

Hope this helps!

-Jay

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by GMATGuruNY » Thu Dec 15, 2016 4:04 am
GMATsid2016 wrote:
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 = b^4 = (integer³)^4 = integer^12.
SUFFICIENT.
Hi GMATGuruNY ,

Everything is clear. can you please explain the above part?

Thanks,

Sid
Statement 2: x = b�, where b is an integer.

In my post above, the following was determined:
b = (a/b)³

Since b is an integer, the second equation above implies that the value in blue must also be an integer, as follows:
b = integer³.

Substituting b = integer³ into x = b�, we get:
x = (integer³)�.

Multiplying the exponents, we get:
x = integer¹².

Thus, x is equal to the 12th power of an integer.
SUFFICIENT.

Some test-takers might find the following approach more straightforward:

Test an EASY CASE.
Test POWERS OF 2.

Statement 1:
x = 2³, 2�, 2�, 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 = 2�, 2�, 2¹²...
If x = 2�, 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 = 2¹�, 2¹�, 2²¹, 2²�...
x = 2¹�, 2²�, 2²�...
The next value common to both lists is 2²� = 4¹², which is the 12th power of an integer.
Implication:
To satisfy both statements, x must be the 12th power of an integer.
SUFFICIENT.

The correct answer is C.
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