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x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which

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GeminiHeat
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x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which [#permalink] New post   17 Mar 2021, 03:04
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x is divisible by 144. If 3√x is an integer, then which of the following is 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
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E
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KarunMendiratta
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x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which [#permalink] New post   17 Mar 2021, 10:56
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GeminiHeat wrote:
x is divisible by 144. If 3√x is an integer, then which of the following is 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


If x is divisible by 144 then, x must have all the factors of it

144=122=4232=2432

So x=2432(k)

Cube-rooting both the sides:

3√x=3√2432(k)

3√x=23√2132(k)

This means k must have two 2s and one 3
k=12

So x=2432(12)=123
Therefore, 3√x=12 which will always be divisible by 4 and 12

Hence, option E
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Re: x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which [#permalink] New post   29 Mar 2021, 22:34
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GeminiHeat wrote:
x is divisible by 144. If 3√x is an integer, then which of the following is 3√x definitely divisible by?

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 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, 3√x≥12. And 12 can be completely divided by 4 and 12. Hence, E is the correct answer.
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Re: x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which [#permalink] New post   24 Mar 2025, 02:57
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Re: x is divisible by 144. If [m]\sqrt[3]{x}[/m] is an integer, then which [#permalink]
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