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.
_________________
GMAT and GRE Tutor
Over 1800 followers
GMATGuruNY@gmail.com
New York, NY
If you find one of my posts helpful, please take a moment to click on the "Kudos" icon.
Available for tutoring in NYC and long-distance.
For more information, please email me at GMATGuruNY@gmail.com.