Find all School-related info fast with the new School-Specific MBA Forum

It is currently 22 Aug 2014, 04:17

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.

Events & Promotions

Events & Promotions in June
Open Detailed Calendar

If the positive integer N is a perfect square, which of the

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Manager
Manager
avatar
Joined: 25 Jul 2010
Posts: 143
Followers: 1

Kudos [?]: 42 [0], given: 29

GMAT Tests User
If the positive integer N is a perfect square, which of the [#permalink] New post 25 Sep 2010, 10:37
3
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

(N/A)

Question Stats:

57% (02:07) correct 43% (01:26) wrong based on 21 sessions
If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

Please write if my understanding is correct.
Expert Post
2 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19047
Followers: 3367

Kudos [?]: 24497 [2] , given: 2680

Re: Perfect square [#permalink] New post 25 Sep 2010, 10:46
2
This post received
KUDOS
Expert's post
Orange08 wrote:
If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

Please write if my understanding is correct.


Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.

Tips about the perfect square:

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.

So I and II must be true.
_________________

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.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

1 KUDOS received
Retired Moderator
User avatar
Joined: 02 Sep 2010
Posts: 807
Location: London
Followers: 77

Kudos [?]: 465 [1] , given: 25

GMAT ToolKit User GMAT Tests User Reviews Badge
Re: Perfect square [#permalink] New post 25 Sep 2010, 10:55
1
This post received
KUDOS
1 is neither considered a prime nor a composite number

The answer to the question should be (1) & (2) only

A perfect square has an odd number of factors
The number of odd factors that a perfect square has is also odd
_________________

Math write-ups
1) Algebra-101 2) Sequences 3) Set combinatorics 4) 3-D geometry

My GMAT story

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Manager
Manager
avatar
Joined: 30 May 2010
Posts: 191
Followers: 3

Kudos [?]: 41 [0], given: 32

GMAT Tests User
Re: Perfect square [#permalink] New post 25 Sep 2010, 12:03
Thanks for the tips Bunuel. I had never thought about 2 and 3, but they make sense.
Intern
Intern
User avatar
Joined: 15 Dec 2009
Posts: 23
Schools: Any offering one year MBA
WE 1: Information Tech
WE 2: Event Consultant for FIFA addi events and WC
Followers: 0

Kudos [?]: 5 [0], given: 14

Re: Perfect square [#permalink] New post 25 Sep 2010, 21:37
On a similar note..
I. The number of factors can be found using the formula = (p+1)(q+1)(s+1).... where p,q,r are the indices

eg1: 25= 5^2 so the number of factors are (2+1) = 3
eg2: 24 = (2^3) * (3^1) so the number of factors are (3+1)(1+1) = 8
_________________

K 4 KUDOS

Manager
Manager
avatar
Joined: 30 May 2010
Posts: 191
Followers: 3

Kudos [?]: 41 [0], given: 32

GMAT Tests User
Re: Perfect square [#permalink] New post 25 Sep 2010, 22:19
g4gmat wrote:
On a similar note..
I. The number of factors can be found using the formula = (p+1)(q+1)(s+1).... where p,q,r are the indices

eg1: 25= 5^2 so the number of factors are (2+1) = 3
eg2: 24 = (2^3) * (3^1) so the number of factors are (3+1)(1+1) = 8


To clarify, this requires the prime factorization of a number.
Manager
Manager
avatar
Joined: 19 Apr 2010
Posts: 216
Schools: ISB, HEC, Said
Followers: 4

Kudos [?]: 17 [0], given: 28

GMAT Tests User
Re: Perfect square [#permalink] New post 26 Sep 2010, 23:05
Nice tips Thanks Bunuel
Senior Manager
Senior Manager
User avatar
Joined: 20 Jan 2010
Posts: 278
Schools: HBS, Stanford, Haas, Ross, Cornell, LBS, INSEAD, Oxford, IESE/IE
Followers: 14

Kudos [?]: 137 [0], given: 117

GMAT Tests User
Re: Perfect square [#permalink] New post 11 Oct 2010, 06:10
Thank you! Bunuel & g4gmat for sharing your tips.
_________________

"Don't be afraid of the space between your dreams and reality. If you can dream it, you can make it so."
Target=780
http://challengemba.blogspot.com
Kudos??

Retired Moderator
User avatar
Status: 2000 posts! I don't know whether I should feel great or sad about it! LOL
Joined: 04 Oct 2009
Posts: 1726
Location: Peru
Schools: Harvard, Stanford, Wharton, MIT & HKS (Government)
WE 1: Economic research
WE 2: Banking
WE 3: Government: Foreign Trade and SMEs
Followers: 65

Kudos [?]: 277 [0], given: 109

GMAT Tests User
Re: Perfect square [#permalink] New post 14 Oct 2010, 05:09
Bunuel wrote:
Orange08 wrote:
If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

Please write if my understanding is correct.


Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.

Tips about the perfect square:

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.

So I and II must be true.


Hi Bunuel!, is there a way of picking numbers to solve this question?
I think that it would be better than trying to remember the rules about perfect squares during the test :?
Thanks!
_________________

"Life’s battle doesn’t always go to stronger or faster men; but sooner or later the man who wins is the one who thinks he can."

My Integrated Reasoning Logbook / Diary: my-ir-logbook-diary-133264.html

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19047
Followers: 3367

Kudos [?]: 24497 [0], given: 2680

Re: Perfect square [#permalink] New post 14 Oct 2010, 05:46
Expert's post
metallicafan wrote:
Bunuel wrote:
Orange08 wrote:
If the positive integer N is a perfect square, which of the following must be true?

I. The number of distinct factors of N is odd.
II. The sum of the distinct factors of N is odd.
III. The number of distinct prime factors of N is even.

For III, 1 is not considered as prime factor.
So, for example, for 4, distinct prime factor would be 2 [highlight]only[/highlight] and not 2 and 1 both.
Likewise, for 9, distinct prime factor would be 3 [highlight]only[/highlight]

Please write if my understanding is correct.


Yes, your understanding of III is right. Prime factor of 4 is 2 and prime factor of 9 is 3. So III is not alway true: a perfect square can have any number of prime factors.

Tips about the perfect square:

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD.
3. A perfect square ALWAYS has an ODD number of Odd-factors, and EVEN number of Even-factors.
4. Perfect square always has even powers of its prime factors.

So I and II must be true.


Hi Bunuel!, is there a way of picking numbers to solve this question?
I think that it would be better than trying to remember the rules about perfect squares during the test :?
Thanks!


Those are useful properties which are worth to remember, even better if you understand why they are right.

Questions about these properties with explanation why they are right:
help-factors-problem-99145.html?hilit=perfect%20square
perfect-square-94700.html?hilit=perfect%20square

As for picking numbers: you can easily prove that III is not always true as soon as you pick appropriate perfect square, say n=2^2=4 --> 4 has 1 (so odd) prime factor, which is 2. For I and II if you try 2-3 perfect squares you'll see that all of them will have the odd number of distinct factors and the odd sum of the distinct factors and though 2-3 examples do not prove that these statement are ALWAYS true you can make educated guess.

The question asks which of the following MUST be true, or which of the following is ALWAYS true no matter what set of numbers you choose. Generally for such kind of questions if you can prove that a statement is NOT true for one particular set of numbers, it will mean that this statement is not always true and hence not a correct answer.

As for "COULD BE TRUE" questions:
The questions asking which of the following COULD be true are different: if you can prove that a statement is true for one particular set of numbers, it will mean that this statement could be true and hence is a correct answer.

Hope it helps.
_________________

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.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Intern
Intern
avatar
Joined: 31 May 2010
Posts: 38
Schools: ESADE, IE
Followers: 0

Kudos [?]: 1 [0], given: 2

Re: Perfect square [#permalink] New post 29 Aug 2011, 09:14
Thank you Bunuel !

I was not aware of 3 and 4. :)
Manager
Manager
avatar
Joined: 09 Jun 2011
Posts: 106
Followers: 0

Kudos [?]: 12 [0], given: 0

Re: Perfect square [#permalink] New post 01 Sep 2011, 19:39
These two facts about perfect square is applicable in this problem, so the Option I and II is true

1. The number of distinct factors of a perfect square is ALWAYS ODD.
2. The sum of distinct factors of a perfect square is ALWAYS ODD
Intern
Intern
avatar
Joined: 01 Feb 2012
Posts: 16
Location: United States
Concentration: Finance, Strategy
GMAT 1: 640 Q49 V28
GMAT 2: 670 Q47 V35
GPA: 3.75
WE: Corporate Finance (Health Care)
Followers: 0

Kudos [?]: 6 [0], given: 4

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 26 Sep 2012, 03:30
Bunuel,

Can you explain pt4 in detail. What if the question had III instead as "prime factors of N are always odd". I think the number prime factor for perfect square will always be odd.
Senior Manager
Senior Manager
avatar
Joined: 16 Dec 2011
Posts: 453
Followers: 9

Kudos [?]: 141 [0], given: 70

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 06 Oct 2012, 19:54
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.
Expert Post
1 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19047
Followers: 3367

Kudos [?]: 24497 [1] , given: 2680

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 07 Oct 2012, 03:54
1
This post received
KUDOS
Expert's post
doe007 wrote:
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.


That's correct:

Tips about perfect squares:
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 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.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Intern
Intern
avatar
Status: Fighting to kill GMAT
Joined: 23 Sep 2012
Posts: 33
Location: United States
Concentration: International Business, General Management
Schools: Duke '16
GPA: 3.8
WE: General Management (Other)
Followers: 0

Kudos [?]: 11 [0], given: 201

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 07 Oct 2012, 05:15
Is 0 considered a perfect square?
_________________

Kudos is the currency of appreciation.



You can have anything you want if you want it badly enough. You can be anything you want to be and do anything you set out to accomplish, if you hold to that desire with the singleness of purpose. ~William Adams

Many of life's failures are people who did not realize how close to success they were when they gave up. ~Thomas A. Edison

Wir müssen wissen, Wir werden wissen. (We must know, we will know.) ~Hilbert

Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19047
Followers: 3367

Kudos [?]: 24497 [0], given: 2680

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 07 Oct 2012, 07:22
Expert's post
closed271 wrote:
Is 0 considered a perfect square?


A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

Since 0=0^2 then 0 is a perfect square. But the properties discussed do not apply to 0.
_________________

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.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Intern
Intern
avatar
Status: Fighting to kill GMAT
Joined: 23 Sep 2012
Posts: 33
Location: United States
Concentration: International Business, General Management
Schools: Duke '16
GPA: 3.8
WE: General Management (Other)
Followers: 0

Kudos [?]: 11 [0], given: 201

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 07 Oct 2012, 08:57
Bunuel wrote:
closed271 wrote:
Is 0 considered a perfect square?


A perfect square, is an integer that is the square of an integer. For example 16=4^2, is a perfect square.

Since 0=0^2 then 0 is a perfect square. But the properties discussed do not apply to 0.


That is what I wanted to point out - that those properties do not apply to 0. Thank you for confirming. However, the question being discussed mentions 'a positive integer', so it should be fine.
_________________

Kudos is the currency of appreciation.



You can have anything you want if you want it badly enough. You can be anything you want to be and do anything you set out to accomplish, if you hold to that desire with the singleness of purpose. ~William Adams

Many of life's failures are people who did not realize how close to success they were when they gave up. ~Thomas A. Edison

Wir müssen wissen, Wir werden wissen. (We must know, we will know.) ~Hilbert

Intern
Intern
avatar
Joined: 21 Oct 2012
Posts: 19
Location: United States
Concentration: Marketing, Operations
GMAT 1: 650 Q42 V36
GPA: 3.6
WE: Information Technology (Computer Software)
Followers: 0

Kudos [?]: 8 [0], given: 15

If the positive integer N is a perfect square, which of the [#permalink] New post 30 Jul 2014, 03:35
Bunuel wrote:
doe007 wrote:
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.


That's correct:

Tips about perfect squares:
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.


Hi Bunuel,

In statement 2 you say that the sum of distinct factors of a perfect square is ALWAYS odd but if we consider the perfect square 49 its factors are =7*7*1 and here the distinct factoros of 49 are 7 and 1 which sum to 8 an EVEN number. Can you please help me understand how statement 2 is always true?

Thanks,
Aamir.
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 19047
Followers: 3367

Kudos [?]: 24497 [0], given: 2680

Re: If the positive integer N is a perfect square, which of the [#permalink] New post 30 Jul 2014, 03:38
Expert's post
havoc7860 wrote:
Bunuel wrote:
doe007 wrote:
With reference to point#2, though "The sum of distinct factors of a perfect square is ALWAYS ODD", the vice versa may not be true. Consider the number 2 (factors 1 & 2) and number 8 (factors 1, 2, 4, & 8) -- these are not perfect squares but sum of their distinct factors are odd.


That's correct:

Tips about perfect squares:
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.


Hi Bunuel,

In statement 2 you say that the sum of distinct factors of a perfect square is ALWAYS odd but if we consider the perfect square 49 its factors are =7*7*1 and here the distinct factoros of 49 are 7 and 1 which sum to 8 an EVEN number. Can you please help me understand how statement 2 is always true?

Thanks,
Aamir.


Factors of 49 are 1, 7, and 49: 1 + 7 + 49 = 57 = odd.
_________________

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.


What are GMAT Club Tests?
25 extra-hard Quant Tests

Get the best GMAT Prep Resources with GMAT Club Premium Membership

Re: If the positive integer N is a perfect square, which of the   [#permalink] 30 Jul 2014, 03:38
    Similar topics Author Replies Last post
Similar
Topics:
Experts publish their posts in the topic Is the positive integer N a perfect square? (1) The number mojorising800 2 10 Jul 2011, 23:26
9 Experts publish their posts in the topic Is the positive integer N a perfect square? PTK 13 23 May 2010, 11:02
11 Experts publish their posts in the topic Is the positive integer N a perfect square? netcaesar 17 13 Aug 2009, 05:49
1 Is the positive integer N a perfect square? (1) The number noboru 2 27 Jul 2009, 14:27
27 Experts publish their posts in the topic Is the positive integer N a perfect square? (1) The number mbaMission 24 02 Jun 2009, 04:18
Display posts from previous: Sort by

If the positive integer N is a perfect square, which of the

  Question banks Downloads My Bookmarks Reviews Important topics  

Go to page    1   2    Next  [ 21 posts ] 



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

Powered by phpBB © phpBB Group and phpBB SEO

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®.