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.

We are given that x is an integer greater than 1 and must determine whether x is equal to the 12th power of an integer.

Statement One Alone:

x is equal to the 3rd power of an integer.

Using the information in statement one, we cannot determine whether x is equal to the 12th power of an integer. For example, if x = 8 = 2^3, then it’s not equal to the 12th power of an integer. However, if x = (2^4)^3 = 2^12, then it is equal to the 12th power of an integer. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

x is equal to the 4th power of an integer

Using the information in statement two, we cannot determine whether x is equal to the 12th power of an integer. For example, if x = 16 = 2^4, then it’s not equal to the 12th power of an integer. However, if x = (2^3)^4 = 2^12, then it is equal to the 12th power of an integer. Statement two alone is not sufficient to answer the question.

Statements One and Two Together:

Using the information from statements one and two, we know that x is equal to the 3rd power of an integer and that x is also equal to the 4th power of some other integer. Let’s represent x as a^3 where a is an integer > 1. Since a^3 is also a 4th power, the 4th root of a^3 is an integer. The only way this could happen is if a is also the 4th power of an integer; in other words, a by itself is a 4th power, say a = b^4 where b is an integer > 1.

Thus, x = a^3 = (b^4)^3 = b^12. Therefore, x is equal to the 12th power of an integer.

Answer: C

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