Practice GMAT Data Sufficiency Question

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If x and y are distinct positive integers, what is the value of x⁴ - y⁴?

(y² + x²)(y + x)(x - y) = 240
x^y = y^x and x > y

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

Before even evaluating the statements, simplify the question. In a more complicated data sufficiency problem, it is likely that some rearranging of the terms will be necessary in order to see the correct answer.
Use the formula for a difference of squares (a² - b²) = (a + b)(a - b). However, let x² equal a, meaning a² = x⁴.
x⁴ - y⁴ = (x² + y²)(x² - y²)
Recognize that the expression contains another difference of squares and can be simplified even further.
(x² + y²)(x² – y²) = (x² + y²)(x – y)(x + y)
The question can now be simplified to: "If x and y are distinct positive integers, what is the value of (x² + y²)(x – y)(x + y)?" If you can find the value of (x² + y²)(x - y)(x + y) or x⁴ - y⁴, you have sufficient data.
Evaluate Statement (1) alone.
1. Statement (1) says (y² + x²)(y + x)(x - y) = 240. The information in Statement (1) matches exactly the simplified question. Statement (1) is SUFFICIENT.
Evaluate Statement (2) alone.
1. Statement (2) says x^y = y^x and x > y. In other words, the product of multiplying x together y times equals the product of multiplying y together x times.
2. The differences in the bases must compensate for the fact that y is being multiplied more times than x (since x > y and y is being multiplied x times while x is being multiplied y times).
3. 4 and 2 are the only numbers that work because only 4 and 2 satisfy the equation n² = 2ⁿ, which is the condition that would be necessary for the equation to hold true.
4. Observe that this is true: 4² = 2⁴ = 16.
5. Remember that x > y, so x = 4 and y = 2. Consequently, you know the value of x⁴ - y⁴ from Statement (2). So, Statement (2) is SUFFICIENT.
Since Statement (1) alone is SUFFICIENT and Statement (2) alone is SUFFICIENT, answer D is correct.

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