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If x is an integer greater than 1, is x equal to the 12th power of an integer?

I. x is equal to the 3rd Power of an integer

II. x is equal to the 4th Power of an integer.

Option A:

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.

Option B:

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.

Option C:

BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.

Option D:

EACH statement ALONE is sufficient to answer the question asked.

Option E:

Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Option C is the correct answer.

Statement I is insufficient: x is equal to the 3rd Power of an integer –> x=m^3 for some positive integer m. If m itself is 4th power of some integer (for example if m=2^4), then the answer will be YES (since in this case x=(2^4)^3=2^12), but if it’s not (for example if m=2), then the answer will be NO. Not sufficient.

(i) Notice that from this statement we have that x^4=m^12. Statement II is insufficient: x is equal to the 4th Power of an integer –> x=n^4 for some positive integer n. If n itself is 3rd power of some integer (for example if n=2^3), then the answer will be YES (since in this case x=(2^3)^4=2^12), but if it’s not (for example if n=2), then the answer will be NO. Not sufficient.

(ii) Notice that from this statement we have that x^3=n^12.

Together I and II,

Divide (i) by (ii): x=(m/n)^12=integer. Now, m/n can be neither an irrational number (since it’s the ratio of two integers) nor some reduced fraction (since no reduced fraction, like 1/2 or 3/2, when raised to some positive integer power can give an integer), therefore m/n must be an integer,

hence x=(m/n)^12=(integer)^12. Sufficient

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