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GRE Prep Club Team Member
Joined: 20 Feb 2017
Posts: 2506
Given Kudos: 1053
GPA: 3.39
x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which
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17 Mar 2021, 03:04
17 Mar 2021, 03:04
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Question Stats:
70% (02:26) correct
29% (02:02) wrong based on 17 sessions
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x is divisible by 144. If \(\sqrt[3]{x}\) is an integer, then which of the following is \(\sqrt[3]{x}\) definitely divisible by?
I. 4
II. 8
III. 12
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
I. 4
II. 8
III. 12
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
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Joined: 16 Apr 2020
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x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which
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17 Mar 2021, 10:56
17 Mar 2021, 10:56
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GeminiHeat wrote:
x is divisible by 144. If \(\sqrt[3]{x}\) is an integer, then which of the following is \(\sqrt[3]{x}\) definitely divisible by?
I. 4
II. 8
III. 12
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
I. 4
II. 8
III. 12
A. I only
B. II only
C. III only
D. I and II only
E. I and III only
If \(x\) is divisible by \(144\) then, \(x\) must have all the factors of it
\(144 = 12^2 = 4^23^2 = 2^43^2\)
So \(x = 2^43^2(k)\)
Cube-rooting both the sides:
\(\sqrt[3]{x} = \sqrt[3]{2^43^2(k)}\)
\(\sqrt[3]{x} = 2\sqrt[3]{2^13^2(k)}\)
This means \(k\) must have two \(2\)s and one \(3\)
\(k = 12\)
So \(x = 2^43^2(12) = 12^3\)
Therefore, \(\sqrt[3]{x} = 12\) which will always be divisible by \(4\) and \(12\)
Hence, option E
Re: x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which
[#permalink]
29 Mar 2021, 22:34
29 Mar 2021, 22:34
3
GeminiHeat wrote:
x is divisible by 144. If \(\sqrt[3]{x}\) is an integer, then which of the following is \(\sqrt[3]{x}\) definitely divisible by?
I. 4
II. 8
III. 12
I. 4
II. 8
III. 12
Since x is divisible by 144, x is a multiple of 144 (a square of 12), and is at least 144. Given \(\sqrt[3]{x}\) is an integer, x must be a perfect cube of an integer. The LEAST perfect cube that x can be is a perfect cube of 12, since we know x is already a perfect square of 12. So, \(\sqrt[3]{x} ≥ 12\). And 12 can be completely divided by 4 and 12. Hence, E is the correct answer.
