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What is the greatest positive integer x such that 9^(6x) is a factor

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Bunuel
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What is the greatest positive integer x such that 9^(6x) is a factor [#permalink] New post   20 Mar 2020, 05:43
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What is the greatest positive integer x such that \(9^{6x}\) is a factor of \(81^{10+x}\)?

A. 2
B. 4
C. 5
D. 6
E. 15
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C
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Kinshook
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Re: What is the greatest positive integer x such that 9^(6x) is a factor [#permalink] New post   20 Mar 2020, 11:22
Bunuel wrote:
What is the greatest positive integer x such that \(9^{6x}\) is a factor of \(81^{10+x}\)?

A. 2
B. 4
C. 5
D. 6
E. 15


Asked: What is the greatest positive integer x such that \(9^{6x}\) is a factor of \(81^{10+x}\)?

\(9^{6x}\) is a factor of \(81^{10+x}\)
\(3^{12x}\) is a factor of \(3^{40+4x}\)
12x <= 40 + 4x
3x <= 10 + x
2x <= 10
x<= 5

IMO C
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Re: What is the greatest positive integer x such that 9^(6x) is a factor [#permalink] New post   21 Mar 2020, 09:05
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Bunuel wrote:
What is the greatest positive integer x such that \(9^{6x}\) is a factor of \(81^{10+x}\)?

A. 2
B. 4
C. 5
D. 6
E. 15


Solution


    • If \(9^{6x}\) is a factor of \(81^{10+x}\)
      o This means, \(\frac{81^{10+x}}{9^{6x}} = I\), where I is a positive integer.
    • Now, Let’s simplify the above expression,
      o \(\frac{81^{10+x}}{9^{6x}} = I\)

      \(⟹\frac{{(9^2)}^{10+x}}{9^{6x}} = I\)

      \(⟹\frac{9^{20 + 2x}}{9^{6x}} = I\)

      \(⟹9^{20+2x -6x} = I\)

      \(⟹9^{20-4x} = I\)
    • Now, for \(9^{20-4x}\) to be an integer, its power must be greater than or equal to zero.
      o Therefore, \(20 -4x ≥0\)
      \(⟹ 5-x ≥0\)
      \(⟹ 5 ≥x\)
    • So, the greatest positive integral value of \(x = 5\).
Thus, the correct answer is Option C.
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Re: What is the greatest positive integer x such that 9^(6x) is a factor [#permalink] New post   31 Mar 2020, 06:02
You can test the answer.

I would go with the middle option first. Let x=5.

[(9^2)^15]/9^30 = (9^30)/(9^30) = 1

Therefore, (C) works.

You can tell (E) is too big. But, let's quickly check (D) to make sure.

Let x=6

[(9^2)^16]/ (9^36) = (9^32)/(9^36) = less than 1!

Therefore, (C) is the correct answer.
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Re: What is the greatest positive integer x such that 9^(6x) is a factor [#permalink] New post   09 Jan 2024, 15:58
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Re: What is the greatest positive integer x such that 9^(6x) is a factor [#permalink]
09 Jan 2024, 15:58
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