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x and y are positive integers satisfying 0 < x - 3y < 1 and b is the decimal portion of (x + 3y)3. What

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x and y are positive integers satisfying 0 < x - 3y < 1 and b is the decimal portion of (x + 3y)3. What

by Max@Math Revolution » Thu Apr 16, 2020 6:54 am

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[GMAT math practice question]

x and y are positive integers satisfying 0 < x - \(\sqrt{3}\) y < 1 and b is the decimal portion of (x + \(\sqrt{3}\) y)^3. What is the value of (x - \(\sqrt{3}\) y)^3 in terms of b?

A. b
B. -b
C. 1 + b
D. 1 - b
E. 3b
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Re: x and y are positive integers satisfying 0 < x - 3y < 1 and b is the decimal portion of (x + 3y)3. What

by Max@Math Revolution » Sun Apr 19, 2020 7:32 pm
=>

Assume a is the decimal portion of (x - \(\sqrt{3}\) y)^3.
Since the integer portion of (x - \(\sqrt{3}\) y)^3 is 0, we have a = (x - \(\sqrt{3}\) y)^3.
If c is the integer portion of (x + \(\sqrt{3}\) y)^3, we have (x + \(\sqrt{3}\) y)^3 = c + b.
Then, we have (x - \(\sqrt{3}\) y)^3 + (x + \(\sqrt{3}\) y)^3 = 2(x^3 + 9xy^2) = a + c + b.
a + b = 2(x^3 + 9xy^2) - c
Since x, y, and c are positive integers, a + b is an integer.
Since we have 0 < a < 1 and 0 ≤ b < 1, we have 0 < a + b < 2.
Thus a + b = 1.
(x - \(\sqrt{3}\) y)^3 = a = 1 - b.

Therefore, D is the answer.
Answer: D
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