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

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Math Expert
Joined: 02 Sep 2009
Posts: 64243
What is the greatest positive integer x such that 9^(6x) is a factor  [#permalink]

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20 Mar 2020, 04:43
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15% (low)

Question Stats:

78% (01:18) correct 22% (01:22) wrong based on 77 sessions

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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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Joined: 03 Jun 2019
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GMAT 1: 690 Q50 V34
WE: Engineering (Transportation)
Re: What is the greatest positive integer x such that 9^(6x) is a factor  [#permalink]

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20 Mar 2020, 10:22
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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Kinshook Chaturvedi
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Re: What is the greatest positive integer x such that 9^(6x) is a factor  [#permalink]

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21 Mar 2020, 08:05
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]

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31 Mar 2020, 05:02

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.
Re: What is the greatest positive integer x such that 9^(6x) is a factor   [#permalink] 31 Mar 2020, 05:02