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# if n and y are positive integers and 450y= n^3. Which of the

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Senior Manager
Joined: 07 Feb 2008
Posts: 301
if n and y are positive integers and 450y= n^3. Which of the [#permalink]

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04 Oct 2008, 12:19
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if n and y are positive integers and 450y= n^3. Which of the following must be an integer.

I) y / 3 x 2^2 x 5

II) y / 3^2 x 2 x 5

III) y / 3 x 2 x 5^2

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Manager
Joined: 27 Sep 2008
Posts: 76
Re: If n and Y .... [#permalink]

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04 Oct 2008, 12:32
450*y= n^3

450 = 5*3*3*2*5

y must be at least 5*3*2*2 so (5*2*3)^3 = n^3

VP
Joined: 30 Jun 2008
Posts: 1018
Re: If n and Y .... [#permalink]

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04 Oct 2008, 13:06
I think the key here is to realize that a cubes have prime factors with powers as 3 or multiples of 3.

eg. 27 = cube of 3 => 3³ ....This has 3 as its power

450y= n^3 ===> this implies that 450y is a cube of some number.

Now prime factors of 450 = 3*3*5*2*5 = 3²*5²*2

given that 450y is a cube => (3²*5²*2)y is a cube.

(3²*5²*2)y will be a cube only when y supplies the remaining number of prime factors to make 450y a cube

i.e. y must contain one 3, one 5 and two 2s ===> 3*5*2² (only this can make y a perfect cube)

so y/3*5*2² must be an integer

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Re: If n and Y ....   [#permalink] 04 Oct 2008, 13:06
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# if n and y are positive integers and 450y= n^3. Which of the

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