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.
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