Practice GMAT Data Sufficiency Question
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If x and y are distinct positive integers, what is the value of x^{4}  y^{4}?
 (y^{2} + x^{2})(y + x)(x  y) = 240
 x^{y} = y^{x} and x > y
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 (a^{2}  b^{2}) = (a + b)(a  b). However, let x^{2} equal a, meaning a^{2} = x^{4}.
x^{4}  y^{4} = (x^{2} + y^{2})(x^{2}  y^{2})  Recognize that the expression contains another difference of squares and can be simplified even further.
(x^{2} + y^{2})(x^{2} – y^{2}) = (x^{2} + y^{2})(x – y)(x + y)  The question can now be simplified to: "If x and y are distinct positive integers, what is the value of (x^{2} + y^{2})(x – y)(x + y)?" If you can find the value of (x^{2} + y^{2})(x  y)(x + y) or x^{4}  y^{4}, you have sufficient data.

Evaluate Statement (1) alone.
 Statement (1) says (y^{2} + x^{2})(y + x)(x  y) = 240. The information in Statement (1) matches exactly the simplified question. Statement (1) is SUFFICIENT.

Evaluate Statement (2) alone.
 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.
 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).
 4 and 2 are the only numbers that work because only 4 and 2 satisfy the equation n^{2} = 2^{n}, which is the condition that would be necessary for the equation to hold true.
 Observe that this is true: 4^{2} = 2^{4} = 16.
 Remember that x > y, so x = 4 and y = 2. Consequently, you know the value of x^{4}  y^{4} 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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