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What is the sum of all possible solutions of the equation 3n-93+(9n-3)3=(3n+9n-12)3, where n is

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What is the sum of all possible solutions of the equation 3n-93+(9n-3)3=(3n+9n-12)3, where n is

by Max@Math Revolution » Fri Apr 24, 2020 1:33 am

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[GMAT math practice question]

What is the sum of all possible solutions of the equation (3^n-9)^3+(9^n-3)^3=(3^n+9^n-12)^3, where n is a positive integer?

A. 1
B. 2
C. 3
D. 4
E. 5
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Re: What is the sum of all possible solutions of the equation 3n-93+(9n-3)3=(3n+9n-12)3, where n is

by Max@Math Revolution » Sun Apr 26, 2020 7:48 pm
=>

Assume a = 3n - 9 and b = 9n - 3.

We have the equation a^3 + b^3 = (a + b)^3 or 3ab(a + b) = 0, since (a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3 = a^3 + b^3 + 3ab(a + b).
Then we have a = 0, b = 0 or a + b = 0.

Case 1: a = 0
Since we have 3^n – 9 = 0, we have n = 2.

Case 2: b = 0
Since we have 9^n – 3 = 0, we have n = ½, which is not a solution, since n is a positive integer.

Case 3: a + b = 0
3^n + 9^n – 12 = 0
=> 9^n + 3^n – 12 = 0
=> 3^2n + 3^n – 12 = 0
=> (3^n - 3)(3^n + 4) = 0
=> 3^n = 3 since 3^n + 4 > 0
=> n = 1

Then, the sum of all solutions is 1 + 2 = 3.

Therefore, C is the answer.
Answer: C
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