It is currently 17 Dec 2017, 04:10

Close

GMAT Club Daily Prep

Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track
Your Progress

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

Not interested in getting valuable practice questions and articles delivered to your email? No problem, unsubscribe here.

Close

Request Expert Reply

Confirm Cancel

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

Is the positive integer N a perfect square? (1) The number

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:

Hide Tags

Director
Director
User avatar
Joined: 22 Mar 2011
Posts: 610

Kudos [?]: 1093 [0], given: 43

WE: Science (Education)
Re: Is the positive integer N a perfect square? [#permalink]

Show Tags

New post 31 Jul 2012, 00:26
ashish8 wrote:
How is B sufficient? Sum of distinct factors of a perfect square is odd, but if n is 2, then the sum is also odd.



(2) states: The sum of all distinct factors of N is even.
Since the sum of distinct factors of a perfect square must be odd, we can conclude that N is not a perfect square.
So, the answer to the question "Is N a perfect square?" is a definite NO.
Therefore, (2) sufficient.

Not only perfect squares have the sum of their distinct factors odd. As you mentioned, for 2, the sum of its factors is odd, and it is not a perfect square.
So, if a number is a perfect square, then the sum of its factors is necessarily odd, but the reciprocal is not true. Meaning, if the sum of the factors is odd, the number is not necessarily a perfect square, it might be or not. But if the sum of the distinct factors is even, then certainly the number cannot be a perfect square.
_________________

PhD in Applied Mathematics
Love GMAT Quant questions and running.

Kudos [?]: 1093 [0], given: 43

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42646

Kudos [?]: 135934 [0], given: 12716

Re: Is the positive integer N a perfect square? [#permalink]

Show Tags

New post 31 Jul 2012, 00:32
ashish8 wrote:
How is B sufficient? Sum of distinct factors of a perfect square is odd, but if n is 2, then the sum is also odd.


Also check this:

Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

Hope it helps.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

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.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 135934 [0], given: 12716

Manager
Manager
avatar
Joined: 22 Jun 2012
Posts: 53

Kudos [?]: 20 [0], given: 6

GMAT 1: 730 Q49 V40
Re: perfect square [#permalink]

Show Tags

New post 31 Jul 2012, 22:19
cipher wrote:
Pkit wrote:
Please help me. This is the question of 700 level from MGMAT CAT1

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.

I have got this question wrong :( , but I would argue with the OA provided by MGMAT.

[Reveal] Spoiler:
OA D
.

Please provide your reasons and explanations. Thank you.



1) The number of distinct factors of N is even.

Suppose N = 4. It has 3 distinct factors: 1, 2 and 4.
Suppose N = 9. It has 3 distinct factors: 1, 3 and 9.
Suppose N = 16. It has 5 distinct factors: 1, 2, 4, 8, and 16.
Suppose N = 64. It has 7 distinct factors: 1, 2, 4, 8, 16, 32, and 64.
But that not the case. In fact, the case is opposite. So it is sufficient because N is not a square.

(2) The sum of all distinct factors of N is even.

If you follow the above pattern, you see 1 is always there. The sum of all distinct factors except 1 of N is even. If you add 1 on the even sum, that odd. So N is not a square.
But that not the case. In fact, the case is opposite. So it is sufficient because N is again not a square.

So D.

PS: A perfect square always have odd number of factors, for e.g a integer \(n\) and its square \(n^2\)

Now, \(n\) will have always have even number of factors, (take any number and you will realise that factors come in pairs), now \(n^2\) will have all factors which \(n\) has + one more which is \(n^2\)

Hi there,

I have a problem with this method. I think it is flawed but luckily works here.
We can see that the two statements should be true for perfect squares, but in no way have we proved that it is not true for non-perfect square.
For instance, getting a few examples of perfect squares and seeing that the sum of the factors is always odd, doesn't imply that summing the factors of a non-perfect square would not be odd...

The only way to properly answer is to know the properties given by Bunuel IMO.

Kudos [?]: 20 [0], given: 6

Manager
Manager
User avatar
Joined: 22 Feb 2009
Posts: 207

Kudos [?]: 184 [0], given: 148

GMAT ToolKit User
Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 21 Aug 2014, 00:26
Bunuel wrote:
tingle15 wrote:
I have a doubt...

Consider N=18, Its factors are: 1, 2, 3, 6, 9, 18. The sum of factors is 39 which is odd... Am i missing something?


Tips about the perfect square:
1. The number of distinct factors of a perfect square is ALWAYS ODD. The reverse is also true: if a number has the odd number of distinct factors then it's a perfect square;

2. The sum of distinct factors of a perfect square is ALWAYS ODD. The reverse is NOT always true: a number may have the odd sum of its distinct factors and not be a perfect square. For example: 2, 8, 18 or 50;

3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors. The reverse is also true: if a number has an ODD number of Odd-factors, and EVEN number of Even-factors then it's a perfect square. For example: odd factors of 36 are 1, 3 and 9 (3 odd factor) and even factors are 2, 4, 6, 12, 18 and 36 (6 even factors);

4. Perfect square always has even powers of its prime factors. The reverse is also true: if a number has even powers of its prime factors then it's a perfect square. For example: \(36=2^2*3^2\), powers of prime factors 2 and 3 are even.

NEXT:
There is a formula for Finding the Number of Factors of an Integer:

First make prime factorization of an integer \(n=a^p*b^q*c^r\), where \(a\), \(b\), and \(c\) are prime factors of \(n\) and \(p\), \(q\), and \(r\) are their powers.

The number of factors of \(n\) will be expressed by the formula \((p+1)(q+1)(r+1)\). NOTE: this will include 1 and n itself.

Example: Finding the number of all factors of 450: \(450=2^1*3^2*5^2\)

Total number of factors of 450 including 1 and 450 itself is \((1+1)*(2+1)*(2+1)=2*3*3=18\) factors.

Back to the original question:

Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even --> let's say \(n=a^p*b^q*c^r\), given that the number of factors of \(n\) is even --> \((p+1)(q+1)(r+1)=even\). But as we concluded if \(n\) is a perfect square then powers of its primes \(p\), \(q\), and \(r\) must be even, and in this case number of factors would be \((p+1)(q+1)(r+1)=(even+1)(even+1)(even+1)=odd*odd*odd=odd\neq{even}\). Hence \(n\) can not be a perfect square. Sufficient.

(2) The sum of all distinct factors of N is even --> if \(n\) is a perfect square then (according to 3) sum of odd factors would be odd and sum of even factors would be even, so sum of all factors of perfect square would be \(odd+even=odd\neq{even}\). Hence \(n\) can not be a perfect square. Sufficient.

Answer: D.

Hope it helps.


Hi, could you explain why " A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors" is true?
Thanks
_________________

.........................................................................
+1 Kudos please, if you like my post

Kudos [?]: 184 [0], given: 148

Expert Post
Veritas Prep GMAT Instructor
User avatar
G
Joined: 16 Oct 2010
Posts: 7799

Kudos [?]: 18148 [0], given: 236

Location: Pune, India
Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 24 Aug 2014, 20:49
vad3tha wrote:

Hi, could you explain why " A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors" is true?
Thanks


Here is a post that explains this: http://www.veritasprep.com/blog/2010/12 ... t-squares/
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Get started with Veritas Prep GMAT On Demand for $199

Veritas Prep Reviews

Kudos [?]: 18148 [0], given: 236

Manager
Manager
avatar
Joined: 22 Aug 2014
Posts: 190

Kudos [?]: 11 [0], given: 49

Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 17 Feb 2015, 07:58
Hi @veritasprepkarishma/@Bunuel,

For 2nd statement if we take
1)4-perfect square-sum of distinct factors is 2 or 4(2*2 or 4*1)
Condition satisfied

2)8-not a perfect square-sum of distinct factors is 2 or 8(2*2*2 or 8*1)
Condition satisfied still not perfect square


Then how can D be the answer?

Kudos [?]: 11 [0], given: 49

Expert Post
Math Expert
User avatar
V
Joined: 02 Sep 2009
Posts: 42646

Kudos [?]: 135934 [0], given: 12716

Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 17 Feb 2015, 08:39
ssriva2 wrote:
Hi @veritasprepkarishma/@Bunuel,

For 2nd statement if we take
1)4-perfect square-sum of distinct factors is 2 or 4(2*2 or 4*1)
Condition satisfied

2)8-not a perfect square-sum of distinct factors is 2 or 8(2*2*2 or 8*1)
Condition satisfied still not perfect square


Then how can D be the answer?


(2) says that the sum of all distinct factors of N is even.

If N = 4, then its factors are 1, 2, and 4 --> the sum = 1 + 2 + 4 = 7 = odd.

If N = 8, then its factors are 1, 2, 4 and 8 --> the sum = 1 + 2 + 4 +8 = 15 = even.
_________________

New to the Math Forum?
Please read this: Ultimate GMAT Quantitative Megathread | All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:
GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

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.


What are GMAT Club Tests?
Extra-hard Quant Tests with Brilliant Analytics

Kudos [?]: 135934 [0], given: 12716

Intern
Intern
avatar
Joined: 14 Jan 2015
Posts: 7

Kudos [?]: [0], given: 0

Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 18 Feb 2015, 10:21
goldeneagle94 wrote:
Interesting Question !!!

A few facts to review:


A perfect sqaure ALWAYS has an ODD number of factors, whose sum is ALWAYS ODD.

A perfect sqaure ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.

Using the above facts, you can conclude that both statements are sufficient to answer the question.


Perfect Square 36:
(6 x 6)
(3 X 3 X 2 X 2)
4 total factors, 2 distinct factors, and sum is even…?

Kudos [?]: [0], given: 0

Expert Post
EMPOWERgmat Instructor
User avatar
P
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 10419

Kudos [?]: 3700 [0], given: 173

Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: 340 Q170 V170
Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 18 Feb 2015, 20:33
Hi Kitzrow,

You have to note the difference between "factors" and "prime factors"

36 has the following FACTORS:
1 and 36
2 and 18
3 and 12
4 and 9
6

So, there are 9 factors and the sum of those factors is 91.

This example matches the prior statements - 36 has an ODD number of factors and the sum of those factors is ODD.

GMAT assassins aren't born, they're made,
Rich
_________________

760+: Learn What GMAT Assassins Do to Score at the Highest Levels
Contact Rich at: Rich.C@empowergmat.com

Rich Cohen

Co-Founder & GMAT Assassin

Special Offer: Save $75 + GMAT Club Tests Free
  Official GMAT Exam Packs + 70 Pt. Improvement Guarantee
www.empowergmat.com/

***********************Select EMPOWERgmat Courses now include ALL 6 Official GMAC CATs!***********************

Kudos [?]: 3700 [0], given: 173

Current Student
avatar
Joined: 06 Oct 2015
Posts: 6

Kudos [?]: [0], given: 6

Schools: Anderson '18 (M)
GMAT 1: 760 Q48 V46
Reviews Badge
Re: Is the positive integer N a perfect square? [#permalink]

Show Tags

New post 02 Nov 2015, 11:12
Question here:

I thought perfect squares always have an even sum of powers of prime factors? For instance, in the example, 36 = 2^2 * 3^2, powers of 2 + 2 = 4. Yet when you look at the total number of factors (2+1) * (2+1) you get 9...and the explanation nulls statement 1 because there are an odd number of factors... Can anyone elaborate on this? Thank you!

Kudos [?]: [0], given: 6

Current Student
avatar
B
Joined: 20 Mar 2014
Posts: 2673

Kudos [?]: 1791 [0], given: 797

Concentration: Finance, Strategy
Schools: Kellogg '18 (M)
GMAT 1: 750 Q49 V44
GPA: 3.7
WE: Engineering (Aerospace and Defense)
GMAT ToolKit User Premium Member Reviews Badge
Re: Is the positive integer N a perfect square? [#permalink]

Show Tags

New post 02 Nov 2015, 11:18
HunterJ wrote:
Question here:

I thought perfect squares always have an even sum of powers of prime factors? For instance, in the example, 36 = 2^2 * 3^2, powers of 2 + 2 = 4. Yet when you look at the total number of factors (2+1) * (2+1) you get 9...and the explanation nulls statement 1 because there are an odd number of factors... Can anyone elaborate on this? Thank you!


You are missing 1 important point.

When you look at number of factors of a perfect square you do ALL factors including 1 and the number itself.

Example, 25 = 5^2 ---> total number of factors = (2+1) =3 , (1,5,25). You can not just add the powers of perfect squares to get the total number of factors.

Statement 1 is sufficient as it gives a straight "no" to the question" is n a perfect square" as all perfect squares will have odd number of total factors.

Hope this helps.

Kudos [?]: 1791 [0], given: 797

Director
Director
avatar
S
Joined: 12 Nov 2016
Posts: 792

Kudos [?]: 39 [0], given: 166

Re: Is the positive integer N a perfect square? (1) The number [#permalink]

Show Tags

New post 26 Sep 2017, 02:42
mbaMission wrote:
Is the positive integer N a perfect square?

(1) The number of distinct factors of N is even.
(2) The sum of all distinct factors of N is even.


Nice explanation Bunuel- we could also arrive at those rules by testing square roots and knowing the distinct factor equation

Statement 1

What Bunuel is demonstrating is that this condition cannot allow a perfect square- the number of distinct factors of a number is found by taking the prime factorization of the number and then adding 1 to all the exponents and then multiplying the product being the number of distinct factors. Notice the numbers 9, 49, 100

3^2 = 9
3^(2+1) = 3 distinct factors: 9, 3 ,1

49= 7^2
7^(2+1) = 3 distinct factors :1,7,49

10^2= 100
5^2 2^2= 100
5^(2+1) 2^(2+1) = 9 distinct factors: 100, 50, 25, 20, 10 , 5, 4 ,2 ,1

So we can clearly see any number with an even number of distinct factors cannot be a perfect square- all perfect squares will have an odd number of distinct factors

suff

Statement 2

We can just try out a few small values such as 25, 9 , 49 - clearly the sum of all distinct factors of any perfect square will be odd

suff


D

Kudos [?]: 39 [0], given: 166

Re: Is the positive integer N a perfect square? (1) The number   [#permalink] 26 Sep 2017, 02:42

Go to page   Previous    1   2   [ 32 posts ] 

Display posts from previous: Sort by

Is the positive integer N a perfect square? (1) The number

  new topic post reply Question banks Downloads My Bookmarks Reviews Important topics  


GMAT Club MBA Forum Home| About| Terms and Conditions| GMAT Club Rules| Contact| Sitemap

Powered by phpBB © phpBB Group | Emoji artwork provided by EmojiOne

Kindly note that the GMAT® test is a registered trademark of the Graduate Management Admission Council®, and this site has neither been reviewed nor endorsed by GMAC®.