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# Is the positive integer n equal to the square of an integer?

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Is the positive integer n equal to the square of an integer? [#permalink]  22 Apr 2012, 00:16
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Is the positive integer n equal to the square of an integer?

(1) For every prime number p, if p is a divisor of n, then so is p^2
(2) $$\sqrt{n}$$ is an integer
[Reveal] Spoiler: OA

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Last edited by Bunuel on 22 Apr 2012, 00:38, edited 1 time in total.
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Re: Is the positive integer n equal to the square [#permalink]  22 Apr 2012, 00:36
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shikhar wrote:
Is the positive integer n equal to the square of an integer?
(1) For every prime number p, if p is a divisor of n, then so is p2.
(2) is an integer.

Is the positive integer n equal to the square of an integer?

Question: is $$n=integer^2$$? So, basically we are asked whether $$n$$ is a perfect square (a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square.).

(1) For every prime number p, if p is a divisor of n, then so is p^2 --> if $$n=2^2$$ then the answer is YES but if $$n=2^3$$ then the answer is NO (notice that in both case prime number 2 as well as 2^2 are divisors of n, so our condition is satisfied). Not sufficient.

(2) $$\sqrt{n}$$ is an integer --> $$\sqrt{n}=integer$$ --> $$n=integer^2$$. Sufficient.

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Re: Is the positive integer n equal to the square [#permalink]  04 May 2013, 23:40
Bunuel wrote:
shikhar wrote:
Is the positive integer n equal to the square of an integer?
(1) For every prime number p, if p is a divisor of n, then so is p2.
(2) is an integer.

Is the positive integer n equal to the square of an integer?

Question: is $$n=integer^2$$? So, basically we are asked whether $$n$$ is a perfect square (a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square.).

(1) For every prime number p, if p is a divisor of n, then so is p^2 --> if $$n=2^2$$ then the answer is YES but if $$n=2^3$$ then the answer is NO (notice that in both case prime number 2 as well as 2^2 are divisors of n, so our condition is satisfied). Not sufficient.

(2) $$\sqrt{n}$$ is an integer --> $$\sqrt{n}=integer$$ --> $$n=integer^2$$. Sufficient.

ST 1-isnt this telling you all the prime factors of n are raised to even powers which makes n a square number-i got wrong can you please re-explain.
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Re: Is the positive integer n equal to the square of an integer? [#permalink]  05 May 2013, 00:34
Hi Ashima,

This is how I verified statement 1.

Take n = 36, p = 2
If p is a divisor of n, then so is p2. Check for p^2 => 2^2 => 4, 36 is divisible by 4. - OK

Take n = 15, p = 3
If p is a divisor of n, then so is p2. Check for p^2 => 3^2 => 9, 15 is not divisible by 9 - Not OK

Hence statement 1 is insufficient. Hope this is OK.

Maybe Bunuel can help us better.

Regards,
Pritish
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Re: Is the positive integer n equal to the square [#permalink]  05 May 2013, 03:11
Expert's post
ashiima86 wrote:
Bunuel wrote:
shikhar wrote:
Is the positive integer n equal to the square of an integer?
(1) For every prime number p, if p is a divisor of n, then so is p2.
(2) is an integer.

Is the positive integer n equal to the square of an integer?

Question: is $$n=integer^2$$? So, basically we are asked whether $$n$$ is a perfect square (a perfect square, is an integer that can be written as the square of some other integer. For example 16=4^2, is a perfect square.).

(1) For every prime number p, if p is a divisor of n, then so is p^2 --> if $$n=2^2$$ then the answer is YES but if $$n=2^3$$ then the answer is NO (notice that in both case prime number 2 as well as 2^2 are divisors of n, so our condition is satisfied). Not sufficient.

(2) $$\sqrt{n}$$ is an integer --> $$\sqrt{n}=integer$$ --> $$n=integer^2$$. Sufficient.

ST 1-isnt this telling you all the prime factors of n are raised to even powers which makes n a square number-i got wrong can you please re-explain.

No, the first statement says that if a prime number p is a factor of n, then so is p^2, which means that the power of p is more than or equal to 2: it could be 2, 3, ... So, n is not necessarily a prefect square. For example, if $$n=2^2$$ then the answer is YES but if $$n=2^3$$ then the answer is NO (notice that in both case prime number 2 as well as 2^2 are divisors of n, so our condition is satisfied).

Hope it's clear.
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Re: Is the positive integer n equal to the square of an integer? [#permalink]  06 May 2013, 12:00
Is the positive integer n equal to the square of an integer?

(1) For every prime number p, if p is a divisor of n, then so is p^2
(2) root n is an integer

From 1 ) If p=4, than 16 is also a factor. Which can qualify n to be a perfect square.But if p=2 than 4 is also a factor. However we can't say if n is square of an integer or not. Hence Insufficient.

2) If root n is an integer -> N has to be the square of an integer. Sufficient.

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Re: Is the positive integer n equal to the square of an integer? [#permalink]  12 Dec 2014, 06:04
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Re: Is the positive integer n equal to the square of an integer?   [#permalink] 12 Dec 2014, 06:04
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# Is the positive integer n equal to the square of an integer?

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