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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] New post 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] New post 22 Apr 2012, 00:36
Expert's post
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.

Answer: B.
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Re: Is the positive integer n equal to the square [#permalink] New post 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.

Answer: B.



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] New post 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] New post 05 May 2013, 03:11
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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.

Answer: B.



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.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: Is the positive integer n equal to the square of an integer? [#permalink] New post 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.

Answer B.
Re: Is the positive integer n equal to the square of an integer?   [#permalink] 06 May 2013, 12:00
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