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
- xy = yx 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 (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.
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Evaluate Statement (1) alone.
- Statement (1) says (y2 + x2)(y + x)(x - y) = 240. The information in Statement (1) matches exactly the simplified question. Statement (1) is SUFFICIENT.
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Evaluate Statement (2) alone.
- Statement (2) says xy = yx 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 n2 = 2n, which is the condition that would be necessary for the equation to hold true.
- Observe that this is true: 42 = 24 = 16.
- 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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