Powers and integers

[Amrabdelnaby]

Hello Pros,

Could you please help me with this?

Thanks

[GMATGuruNY]

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�, then x = (2�)³ = 2¹², which is the 12th power of an integer.
INSUFFICIENT.

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

Statements 1 and 2 combined:
Since x = a³ and x = b�, we get:
a³ = b�
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:
b = (a/b)³ = (integer)³.

Since b = (integer)³, and x = b�, we get:
x = b� = (integer³)� = (integer)¹².
SUFFICIENT.

The correct answer is C.

[ceilidh.erickson]

Here's another way to think about it:

If X is equal to the 3rd power of an integer, that means we can express it as some integer raised to an exponent that is a multiple of 3. As Mitch showed, a 3rd power could be simplified to 2^3 or 2^12, but either way the exponent must be a multiple of 3.

For statement 2, we can translate similarly: the exponent must be a multiple of 4.

Individually, these statements are not sufficient. But if we put them together, X must be expressible as an integer raised to an exponent that is both a multiple of 3 and a multiple of 4, it must also be a multiple of 12. Sufficient.