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# If n and y are positive integers and 450y=n^3,

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If n and y are positive integers and 450y=n^3, [#permalink]  02 Jan 2013, 19:59
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Difficulty:

55% (hard)

Question Stats:

55% (01:58) correct 45% (01:31) wrong based on 51 sessions
If n and y are positive integers and 450y=n^3, which of the following must be an integer?

I. (y)/{3×(2^2)×5}

II. (y)/{(3^2)×2×5}

III. (y)/{3×2×(5^2)}

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-n-and-y-are-positive-integers-and-450y-n-92562.html
[Reveal] Spoiler: OA
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Re: If n and y are positive integers and 450y=n^3, [#permalink]  02 Jan 2013, 21:00
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Expert's post
kiyo0610 wrote:
If n and y are positive integers and 450y=n^3, which of the following must be an integer?

I. (y)/{3×(2^2)×5}

II. (y)/{(3^2)×2×5}

III. (y)/{3×2×(5^2)}

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

Defactorize 450.
It will come out as $$5^2 * 3^2 * 2$$.
Since n and y are integers, hence $$450y$$ will also be an integer.
To make y an integer 450 must be a factor of $$n^3$$. In order for this to happen, 5s, 3s and 2s must be in trios. When one takes the cube root, the resultant will be an integer.

Only I provides those numbers that make the 5s, 3s and 2s to become in trios. Since 450 = $$5^2 * 3^2 * 2$$, so cube root of the number that makes the resultant root an integer will be the answer.
Only I does that.
Hence B
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Re: If n and y are positive integers and 450y=n^3, [#permalink]  02 Jan 2013, 22:24
n3 = 450 y
n3 = 9 x 5 x 10 y
n3 = 3 x 3 x 5 x 5 x 2 y
Because n is in its 3rd power.
This implies that
Y has to have a 3, a 5, two 2’s and may be other
(lets call it x3) as its prime factors
Thus,
Y = 3 * 5 * 2 * 2 * x3

Condn 1->y/ 3*4*5
Replacing Y with its factors
(3 * 5 * 2 * 2 * x3) / 3*4*5
Nothing left in the denominator -> Must be Integer
I is correct.

[b]Condn 2->[/b]
y/ 3*3*2*5
Replacing Y with its factors
(3 * 5 * 2 * 2 * x3) / 3*3*2*5
A 3 left in the denominator -> May be Integer
II is not correct.

Condn 3->
y/ 3*2*5*5
Replacing Y with its factors
(3 * 5 * 2 * 2 * x3) / 3*2*5*5
A 5 left in the denominator -> May be Integer III is not correct.

Ans is B
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Re: If n and y are positive integers and 450y=n^3, [#permalink]  03 Jan 2013, 01:36
Expert's post
kiyo0610 wrote:
If n and y are positive integers and 450y=n^3, which of the following must be an integer?

I. (y)/{3×(2^2)×5}

II. (y)/{(3^2)×2×5}

III. (y)/{3×2×(5^2)}

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

OPEN DISCUSSION OF THIS QUESTION IS HERE: if-n-and-y-are-positive-integers-and-450y-n-92562.html
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Re: If n and y are positive integers and 450y=n^3,   [#permalink] 03 Jan 2013, 01:36
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