bhupi 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.

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.

Alternate approach:

Statement 1: x = a³, where a is an integer[/b]

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 integerIf 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.

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