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x^3 + y^3 = A(x + y), x^2 + y^2 = B, where x and y are positive integers. What is the value

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by Max@Math Revolution » Fri Apr 24, 2020 1:29 am

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[GMAT math practice question]

x^3 + y^3 = A(x + y), x^2 + y^2 = B, where x and y are positive integers. What is the value of xy?

1) A - B = 36.
2) B = 97.
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Source: — Data Sufficiency |
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Re: x^3 + y^3 = A(x + y), x^2 + y^2 = B, where x and y are positive integers. What is the value

by Max@Math Revolution » Sun Apr 26, 2020 7:45 pm
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Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.
Visit https://www.mathrevolution.com/gmat/lesson for details.

The first step of the VA (Variable Approach) method is to modify the original condition and the question. We then recheck the question. We should simplify conditions if necessary.

We can simplify the equation x^3 + y^3 = A(x + y) found in the original equation as follows:
x^3 + y^3 = A(x + y)
⇒ (x + y)(x^2 – xy + y^2) = A(x + y) (factoring using the sum of cubes)
⇒ [(x + y)(x^2 – xy + y^2)] / (x + y) = A (dividing both sides by (x + y))
⇒ (x^2 – xy + y^2) = A (simplifying)

Since xy = x^2+xy+y^2 – (x^2+y^2) = (x^3+y^3)/(x+y) – (x^2+y^2) = A – B = 36, condition 1) is sufficient.

Condition 2)
Since x^2 + y^2 = 97, the possible pairs of (x, y) are (4, 9) and (9, 4).
If x = 4, y = 9, then we have xy = 36.
If x = 9, y = 4, then we have xy = 36.

Since condition 2) yields a unique solution, it is sufficient.

Therefore, D is the answer.
Answer: D

This question is a CMT 4(B) question: condition 1) is easy to work with, and condition 2) is difficult to work with. For CMT 4(B) questions, D is most likely the answer, since this question is an integer question.
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