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If x and y are positive integers and 180x=y^3, which of the following

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Joined: 30 Aug 2010
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If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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New post 04 Sep 2010, 11:34
5
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A
B
C
D
E

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Question Stats:

69% (01:40) correct 31% (02:09) wrong based on 678 sessions

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If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. \(\frac{x}{(2^2*3*5)}\)

II. \(\frac{x}{(2*3^2*5)}\)

III. \(\frac{x}{(2*3*5^2)}\)


(A) None
(B) I only
(C) II only
(D) III only
(E) I and II
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Joined: 02 Sep 2009
Posts: 55709
Re: If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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New post 04 Sep 2010, 12:01
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rraggio wrote:
If x and y are positive integers and 180x=y^3, which of the following must be an integer?

I. x/(2^2*3*5)
II. x/(2*3^2*5)
III. x/(2*3*5^2)

(A) None
(B) I only
(C) II only
(D) III only
(E) I and II


\(x\) and \(y\) are integers and \(180x=y^3\) --> \(180x\) equals to cube of an integer (\(y^3\)). \(180x=2^2*3^2*5*x=y^3\). The smallest value of \(x\) for which \(2^2*3^2*5*x\) is a cube of an integer is when \(x=2*3*5^2\) (to complete powers of 2, 3 and 5 to cubes). In this case \(180x=(2^2*3^2*5)*(2*3*5^2)=(2*3*5)^3=cube \ of \ an \ integer\). So \(x\) is definitely divisible by \(2*3*5^2\).

Answer: D.

Similar problems:
number-properties-92562.html?hilit=following%20must%20none#p712584
og-quantitative-91750.html#p704028
division-factor-88388.html#p666722

Hope it helps.
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If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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New post 15 May 2018, 10:46
1
rraggio wrote:
If x and y are positive integers and 180x = y^3, which of the following must be an integer?

I. \(\frac{x}{(2^2*3*5)}\)

II. \(\frac{x}{(2*3^2*5)}\)

III. \(\frac{x}{(2*3*5^2)}\)


(A) None
(B) I only
(C) II only
(D) III only
(E) I and II


180=5*6*6
let x=5*5*6 to produce (5*6)^3=y^3
5*5*6=150
only III is an integer
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Re: If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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New post 23 May 2019, 17:54
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Re: If x and y are positive integers and 180x=y^3, which of the following   [#permalink] 23 May 2019, 17:54
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