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

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Intern
Joined: 30 Aug 2010
Posts: 13
If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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04 Sep 2010, 11:34
5
28
00:00

Difficulty:

35% (medium)

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
Math Expert
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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04 Sep 2010, 12:01
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14
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 $$2*3*5^2$$.

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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##### General Discussion
VP
Joined: 07 Dec 2014
Posts: 1196
If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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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
D
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Joined: 09 Sep 2013
Posts: 11394
Re: If x and y are positive integers and 180x=y^3, which of the following  [#permalink]

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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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