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What is the sum of all possible solutions of the equation

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Bunuel
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What is the sum of all possible solutions of the equation [#permalink] New post   26 Jun 2023, 03:00
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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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C
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What is the sum of all possible solutions of the equation [#permalink] New post   26 Jun 2023, 04:22
Bunuel wrote:
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. \(5\)


\((a + b)^3 = a^3 + b^3 + 3ab(a+b)\)

\(a = 3^n-9\)

\(b = 9^n-3\)

\((3^n+9^n-12)^3\) ⇒

\((3^n-9+9^n-3)^3 = (3^n-9)^3 + (9^n-3)^3 + 3(3^n-9)(9^n-3)(3^n-9+9^n-3)\)

\((3^n-9)^3+(9^n-3)^3 = (3^n-9)^3 + (9^n-3)^3 + 3(3^n-9)(9^n-3)(3^n-9+9^n-3)\)

Cancelling \((3^n-9)^3+(9^n-3)^3\)

\(3(3^n-9)(9^n-3)(3^n+9^n-12) = 0\)

Possible Values

  • \((3^n-9) = 0\)
    ⇒ \(n = 2\)
  • \((9^n-3) = 0\)
    ⇒ \(n = \frac{1}{2}\)
  • \((3^n+9^n-12) = 0\)
    \((3^n+9^n) = 12\)
    ⇒ \(n = 1\)

As n is a positive integer, we can discard the value of \(\frac{1}{2}\).

Sum = 2 + 1 = 3

Option C
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Re: What is the sum of all possible solutions of the equation [#permalink] New post   27 Jun 2023, 01:17
Is it better to just plug in values in such type of questions? I can see max sum is 5 according to options so just need to check for n = 1,2 & 3

Option C
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Re: What is the sum of all possible solutions of the equation [#permalink] New post   13 Jul 2023, 09:11
I read somewhere that the GMAT does not expect us to know a lot of formulas and does not expect such complicated expansions for 3 different items. Then I looked at the Q again and I figured 1 could definitely be a solution. Which led me to look at the choices and the choices are carefully given so we can easily find other solutions.

Hence, no need to expand. Simply use plug in.
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Re: What is the sum of all possible solutions of the equation [#permalink]
13 Jul 2023, 09:11
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