richachampion 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. We can eliminate answer choices A and D.
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. We can eliminate answer choice B.
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 fourth power, the fourth root of a^3 is an integer. The only way this could happen is if a is also the fourth power of an integer; in other words, a by itself is a fourth 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
