Question: If x and y are negative integers, then what is the value of xy?

1. \(x^y=\frac{1}{81}\)
2. \(y^x=-\frac{1}{64}\)

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.

Answer:
Solution with Explanation:
Approach Solution (1):

(1) Prime factorization of 81 = 3 * 3 * 3 * 3 = \(3^4 or 9^2 or 81^1 \)

So x could be -3 and y could be -4 or x could be -9 and y could be -2
Insufficient

(2) Prime factorization of 64 = 2 * 2 * 2 * 2 * 2 * 2 = \(2^6 or 4^3 or 8^2 or 64^1\)

So y could be -4 and x could be -3 or y could be -64 and x could be -1
Insufficient

By combining both:
There is only one common pair of x and y (x = -3 and y = -4)

Correct Option: C

Approach Solution (2):

(1) \(x^y=\frac{1}{81}\). As both x and y are negative integers then \(x^y=\frac{1}{81}=(-9)^{-2}=(-3)^{-4}\)

Hence xy = 18 or xy = 12.
Note that as the negative integer (x) in negative integer power (y) gives a positive number, then the power must be a negative even number.
Not sufficient

(2) \(y^x=-\frac{1}{64}\). As a result is negative then x must be negative odd number, so: \(y^x=-\frac{1}{64}=(-4)^{-3}=(-64)^{-1}\) hence xy = 12 or xy = 64

Not sufficient
(1) + (2) only one pair of negative integers x and y satisfies both statements x = -3 and y = -4
Therefore xy = 12
Sufficient

Correct Option: C

“If x and y are negative integers, then what is the value of xy?”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

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