Practice GMAT Data Sufficiency Question

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If n and k are integers and (-2)n⁵ > 0, is k³⁷ < 0?

(nk)^z > 0, where z is an integer that is not divisible by two
k < n

A)	Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
B)	Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
C)	BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D)	EACH statement ALONE is sufficient.
E)	Statement (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data are needed.

Correct Answer: D

It is important to begin by simplifying the question.
Since k is raised to an odd power, k³⁷ will always be less than 0 if k is less than 0. Likewise, k³⁷ will always be greater than 0 if k is greater than 0.
So, the question can be simplified to: is k < 0?
k^{(odd integer)} < 0 if k < 0
k^{(odd integer)} > 0 if k > 0
The question can be simplified even more. Since (-2)(negative number) > 0 and (-2)(positive number) < 0, you know n⁵ is a negative number. This means that n < 0. If n were greater than 0, the statement (-2)n⁵ > 0 would never be true.
Summarizing in algebra:
(-2)(negative number) > 0
(-2)(positive number) < 0
(-2)(n⁵) > 0
n⁵ < 0
Therefore: n < 0
The fully simplified question is: "if n and k are integers and n < 0, is k < 0?"
Evaluate Statement (1) alone.
1. By saying that "z is an integer that is not divisible by 2," Statement (1) is saying that z is an odd integer. So, any base raised to z will keep its sign (i.e., whether the expression is positive or negative will not change since the base is raised to an odd exponent).
  z/2 = not integer if z is odd
  z/2 = integer if z is even
2. Remember that (nk)^z = (n^z)(k^z). So, Statement (1) says that (n^z)(k^z) > 0. It is important to know that there are two ways that a product of two numbers can be greater than zero:
  Case 1: (negative number)(negative number) > 0
  Case 2: (positive number)(positive number) > 0
3. Since you know that n < 0, we are dealing with Case 1 and Statement (1) can be simplified even further:
  (negative number)(k^{(odd exponent)}) > 0.
4. Since k will not change its sign when raised to an odd exponent, the equation can be simplified even further:
  (negative number)(k) > 0. k must be a negative number. Otherwise, this inequality will not be true.
5. To summarize in algebra:
  (nk)^z > 0
  (nk)^z = (n^z)(k^z)
  (n^z)(k^z) > 0
  (negative number)(negative number) > 0
  or (positive number)(positive number) > 0
  (negative number)(k^{(odd exponent)}) > 0
  (negative number)(k) > 0
  k is negative
6. Since k is a negative number, k³⁷ < 0. Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
1. Statement (2) says that k is less than n. Since you know that n is less than 0, Statement (2) says that k is less than a negative number. Only a negative number is less than another negative number. So, k must also be a negative number. Consequently, k³⁷ will always be less than 0 since (negative)^odd < 0. Statement (2) is SUFFICIENT.
2. Summarizing in algebra:
  k < n < 0
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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