Question: If m and n are non-zero integers, is m^n an integer?

(1) n^m is positive
(2) n^m is an integer

1. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
2. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
3. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
4. EACH statement ALONE is sufficient.
5. Statements (1) and (2) TOGETHER are not sufficient.

“If m and n are nonzero integers, is m^n an integer? GMAT Data Sufficiency” - is a topic of the GMAT Quantitative reasoning section of GMAT. The GMAT Quant section includes 31 multiple-choice questions that need to be finished in 62 minutes.GMAT Data Sufficiency questions are followed by a problem statement that consists of two factual statements. This specific GMAT data sufficiency question evaluates the candidate’s mastery of solving calculative mathematical problems. The challenging portion of this type of question mainly comes from clever wording of the question that candidates overlook. GMAT data sufficiency contains 15 questions, two-fifths of the entire 31 GMAT quant questions.

Solutions and Explanation

Approach Solution : 1

Any value of m will be an integer if n is a positive integer (taking into account that both are nonzero integers).

When n is negative, m^n will only be an integer if and only if m=1 or m=-1, as in (-1)^(-2)=1/(-1)^2=1).

In essence, the question is: Is n positive or does m = |1|?

(1) If n^m is positive, it means that either m is even (in this case m can take any value) or n is positive (in this case m can take any value). It's insufficient.
(2) The integer n^m has two possible values: m=positive (in which case n can take any value) or m=negative (in which case n can take either 1 or -1). It's insufficient.

(1)+(2) The answer will be NO if n^m=(-1)^2=positive integer, as m^n=2^(-1)=1/2.
However, the answer will be YES if n^m=1^2=positive integer as m^n=2^1=2.
It's insufficient.

Correct Answer: E

Approach Solution : 2

It is mentioned that m, n are nonzero integers. So m, n are not equal to zero

(1) n^m is positive

n = 1, m = 1, n^m = 1, so m^n = 1. The response we got is YES
n = -1, m = 2, n^m = 1, so m^n = ½. The response we got is NO

This statement is not Sufficient.

(2) n^m is an integer

n = 1, m = 1, n^m = 1, so m^n = 1. The response we got is YES
n = -1, m = 2, n^m = 1, so m^n = ½. The response we got is NO

This statement is not Sufficient.

By combining both these statements, we can use the same examples. But combing is also not sufficient.

Correct Answer: E

Approach Solution : 3

(1) n^m is positive.

Since it is given that n^m is positive, it is clear that n^m>0.

Two possibles cenarios:

(a) By considering n < 0, m should be even
(b) By considering n > 0, then m can be any number

From these 2 scenarios, we can understand that n can take up negative values.
As a result m^n will not be an integer.

This statement is insufficient

(2) n^m is an integer

This states that m>= 0 but n can take up any value.

Again since n can be (+) or (-).

As a result this statement is insufficient

Even by combining the statements, there will not be a difference, n can still take up negative values.

As a result, this is insufficient

Correct Answer: E

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