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# If q is a positive integer, is p*q/q^(1/2) an integer?

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Manager
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If q is a positive integer, is p*q/q^(1/2) an integer? [#permalink]

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27 Apr 2011, 19:57
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If $$q$$ is a positive integer, is $$p\frac{q}{\sqrt{q}}$$ an integer?

(1) $$q = p^2$$

(2) $$p$$ is a positive integer

M02-20

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[Reveal] Spoiler: OA

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Last edited by Bunuel on 02 Dec 2017, 05:26, edited 2 times in total.
Renamed the topic and edited the question.

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Intern
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Re: If q is a positive integer, is p*q/q^(1/2) an integer? [#permalink]

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16 Nov 2014, 21:20
1) q=p^2
take the square root on both sides
p=(sqroot)q

so p*(q/(sqroot)q) = (sqroot)q * q/(sqroot)q = q
and from definition we know that q is a positive integer. sufficient

2)p is an integer
let p=2 and q=3 then the equation becomes 2 * 3 / (sqroot)3 which is not an integer
let p=2 and q=4 then the equations becomes 2 * 4 / (sqroot)4 = 4 which is an integer insufficient

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Math Expert
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Re: If q is a positive integer, is p*q/q^(1/2) an integer? [#permalink]

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17 Nov 2014, 00:39
If $$q$$ is a positive integer, is $$p\frac{q}{\sqrt{q}}$$ an integer?

(1) $$q = p^2$$. Take the square root from botth sides: $$p=\sqrt{q}$$. Substitute $$p$$: $$p\frac{q}{\sqrt{q}}=\sqrt{q}*\frac{q}{\sqrt{q}} = q = \text{integer}$$. Sufficient.

(2) $$p$$ is a positive integer. So, we have that $$p\frac{q}{\sqrt{q}}=\text{integer}*\sqrt{q}$$. This product may or may not be an integer depending on $$q$$. Not sufficient.

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Manager
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Re: If q is a positive integer, is p*q/q^(1/2) an integer? [#permalink]

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17 Nov 2014, 03:25
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Expression given pq/q^1/2

this can be further simplified as (pq/q^1/2)*(q^1/2)/q^1/2

this will reduced to pq^1/2

1)q=p^2
since q is positive integer
the main expression will be =p^2. is an integer. Sufficient
2)p is positive integer.
this does not provide any solution. Insufficient

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Manager
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Re: If q is a positive integer, is p*q/q^(1/2) an integer? [#permalink]

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17 Nov 2014, 10:29
statement 1: sufficient
If q=p^2 and q is a positive integer, then p must also be an integer. After plugging in p^2 for Q in the question , then after reducing you see that it is equal to p^2.

statement 2: insufficient
If q=9 and p=2 then plug into the equation to get 6 as your answer. However, if you plug in q=6 and p=2, the equation does not equal an integer. not enough info.

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Senior Manager
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Re: If q is a positive integer, is p*q/q^(1/2) an integer? [#permalink]

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02 Dec 2017, 05:21
In A - q = p^2 ,
=> sqroot(q) = sqroot(p^2); sqroot(q) = p

Question = p* q / sqroot(q) = p * q / p = q.

It is given q is positive hence it is positive.

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Re: If q is a positive integer, is p*q/q^(1/2) an integer?   [#permalink] 02 Dec 2017, 05:21
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