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

It is currently 01 Sep 2014, 06:08

Starting Soon:

Live Q&A Session with ISB Admissions Team  | |  Join Chat Room to attend this event


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 x is an integer, what is the sum of all distinct positive

  Question banks Downloads My Bookmarks Reviews Important topics  
Author Message
TAGS:
Intern
Intern
avatar
Joined: 18 Feb 2010
Posts: 29
Followers: 1

Kudos [?]: 13 [0], given: 5

GMAT Tests User
If x is an integer, what is the sum of all distinct positive [#permalink] New post 26 Nov 2010, 18:55
5
This post was
BOOKMARKED
00:00
A
B
C
D
E

Difficulty:

  55% (hard)

Question Stats:

49% (02:53) correct 51% (01:56) wrong based on 143 sessions
If x is an integer, what is the sum of all distinct positive factors of \sqrt{x}?

(1) x has exactly 3 distinct positive factors
(2) x^2 -1 =3k where k is an odd integer

Please explain. I could solve this but it took me more than 3 mins.

Thanks
NAD
[Reveal] Spoiler: OA
Expert Post
1 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4691
Location: Pune, India
Followers: 1089

Kudos [?]: 4882 [1] , given: 163

Re: Sum of the distinct factors of square root of a number [#permalink] New post 26 Nov 2010, 20:08
1
This post received
KUDOS
Expert's post
nades09 wrote:
If x is an integer, what is the sum of all distinct positive factors of \sqrt{x}?

(1) x has exactly 3 distinct positive factors
(2) x^2 -1 =3k where k is an odd integer

Please explain. I could solve this but it took me more than 3 mins.

Thanks
NAD


You can quickly solve it using logic. I will give you a teaser and see if you can arrive at the answer on your own.

Statement 1 tells you \sqrt{x} is prime.
Statement 2 tells you x^2 is even.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

2 KUDOS received
Intern
Intern
avatar
Joined: 18 Feb 2010
Posts: 29
Followers: 1

Kudos [?]: 13 [2] , given: 5

GMAT Tests User
Re: Sum of the distinct factors of square root of a number [#permalink] New post 26 Nov 2010, 20:32
2
This post received
KUDOS
Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\sqrt{x}.
Since \sqrt{x} for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) x^2-1 = 3k where k =odd integer

Hence, x^2 =3k+1 is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\sqrt{x} is 2, hence we can find the sum

Ans: C
Expert Post
5 KUDOS received
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4691
Location: Pune, India
Followers: 1089

Kudos [?]: 4882 [5] , given: 163

Re: Sum of the distinct factors of square root of a number [#permalink] New post 27 Nov 2010, 11:16
5
This post received
KUDOS
Expert's post
1
This post was
BOOKMARKED
nades09 wrote:
Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\sqrt{x}.
Since \sqrt{x} for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) x^2-1 = 3k where k =odd integer

Hence, x^2 =3k+1 is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\sqrt{x} is 2, hence we can find the sum

Ans: C


You got most of it. If you go through factors theory, it will help you understand that only a square of a prime number can have 3 factors. e.g. 4 or 25 or 49 or 121 or 169..
1 , 2, 4 are factors of 4
1, 5, 25 are factors of 25
1, 7, 49 are factors of 49
1, 11, 121 are factors of 121 etc

So if x has 3 factors, \sqrt{x} must be prime.

Also from statement 2, x^2 = 3k + 1 where k is odd. So 3k is odd and 3k + 1 is even. So x^2 is even. Now, if x^2 is even, x has to be even too (It is not possible that a power of an odd number becomes even. If x is odd, x^2, x^3 etc all will be odd. If x is even, x^2, x^3etc all will be even.). Then \sqrt{x} must also be even.

The only number that is even and prime is 2. So \sqrt{x} must be 2.
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

SVP
SVP
User avatar
Joined: 09 Sep 2013
Posts: 2231
Followers: 186

Kudos [?]: 37 [0], given: 0

Premium Member
Re: If x is an integer, what is the sum of all distinct positive [#permalink] New post 12 Sep 2013, 08:26
Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.
_________________

GMAT Books | GMAT Club Tests | Best Prices on GMAT Courses | GMAT Mobile App | Math Resources | Verbal Resources

Manager
Manager
avatar
Joined: 17 Mar 2014
Posts: 51
Followers: 0

Kudos [?]: 17 [0], given: 30

If x is an integer, what is the sum of all distinct positive [#permalink] New post 25 Jul 2014, 05:52
VeritasPrepKarishma wrote:
nades09 wrote:
Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\sqrt{x}.
Since \sqrt{x} for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) x^2-1 = 3k where k =odd integer

Hence, x^2 =3k+1 is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\sqrt{x} is 2, hence we can find the sum

Ans: C


You got most of it. If you go through factors theory, it will help you understand that only a square of a prime number can have 3 factors. e.g. 4 or 25 or 49 or 121 or 169..
1 , 2, 4 are factors of 4
1, 5, 25 are factors of 25
1, 7, 49 are factors of 49
1, 11, 121 are factors of 121 etc

So if x has 3 factors, \sqrt{x} must be prime.

Also from statement 2, x^2 = 3k + 1 where k is odd. So 3k is odd and 3k + 1 is even. So x^2 is even. Now, if x^2 is even, x has to be even too (It is not possible that a power of an odd number becomes even. If x is odd, x^2, x^3 etc all will be odd. If x is even, x^2, x^3etc all will be even.). Then \sqrt{x} must also be even.

The only number that is even and prime is 2. So \sqrt{x} must be 2.


I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?
1 KUDOS received
Intern
Intern
User avatar
Joined: 08 Jul 2010
Posts: 5
GMAT: INSIGHT
Followers: 0

Kudos [?]: 6 [1] , given: 1

If x is an integer, what is the sum of all distinct positive [#permalink] New post 25 Jul 2014, 09:40
1
This post received
KUDOS
Quote:
I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?


Factors are not essentially Integers otherwise the expressions of "HCF (Highest Common Factor) of Fractions" would not have existed.

In order to ensure that x is a perfect square we will have to combine the information of first statement with second statement. I hope this will give you an insight why answer option C is the correct answer.
_________________

Regards,
Bhoopendra Singh & Sushma Jha
"GMATinsight"

Mob: +91-9999687183
Mob: +91-9891333772

Manager
Manager
avatar
Joined: 17 Mar 2014
Posts: 51
Followers: 0

Kudos [?]: 17 [0], given: 30

If x is an integer, what is the sum of all distinct positive [#permalink] New post 25 Jul 2014, 15:12
GMATinsight wrote:
Quote:
I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?


Factors are not essentially Integers otherwise the expressions of "HCF (Highest Common Factor) of Fractions" would not have existed.

In order to ensure that x is a perfect square we will have to combine the information of first statement with second statement. I hope this will give you an insight why answer option C is the correct answer.


well if non integers can have factors then the answer is C , Karishma if you can add little more to this that would surely help.

Last edited by qlx on 26 Jul 2014, 01:36, edited 1 time in total.
Expert Post
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 25228
Followers: 3428

Kudos [?]: 25198 [0], given: 2702

Re: If x is an integer, what is the sum of all distinct positive [#permalink] New post 25 Jul 2014, 15:32
Expert's post
qlx wrote:
VeritasPrepKarishma wrote:
nades09 wrote:
Thanks!

(1) The only squares that have 3 distinct positive factors are 4, 25 and 49.
The factors are 1,x,\sqrt{x}.
Since \sqrt{x} for the above three numbers needs to be considered, hence there will be three different values for the sums

Hence (1) is insufficient

(2) x^2-1 = 3k where k =odd integer

Hence, x^2 =3k+1 is even

There can be multiple values where x^2 is even

Hence (2) is insufficient

(1)+(2) - x^2 should be even and should have 3 distinct positive factors
The only value that satisfies both conditions is 4

\sqrt{x} is 2, hence we can find the sum

Ans: C


You got most of it. If you go through factors theory, it will help you understand that only a square of a prime number can have 3 factors. e.g. 4 or 25 or 49 or 121 or 169..
1 , 2, 4 are factors of 4
1, 5, 25 are factors of 25
1, 7, 49 are factors of 49
1, 11, 121 are factors of 121 etc

So if x has 3 factors, \sqrt{x} must be prime.

Also from statement 2, x^2 = 3k + 1 where k is odd. So 3k is odd and 3k + 1 is even. So x^2 is even. Now, if x^2 is even, x has to be even too (It is not possible that a power of an odd number becomes even. If x is odd, x^2, x^3 etc all will be odd. If x is even, x^2, x^3etc all will be even.). Then \sqrt{x} must also be even.

The only number that is even and prime is 2. So \sqrt{x} must be 2.


I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?


Here is another value: x^2 = 3*85 + 1 = 256 --> x = 16 --> \sqrt{x}=4.
_________________

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

Expert Post
1 KUDOS received
Math Expert
User avatar
Joined: 02 Sep 2009
Posts: 25228
Followers: 3428

Kudos [?]: 25198 [1] , given: 2702

Re: If x is an integer, what is the sum of all distinct positive [#permalink] New post 25 Jul 2014, 15:34
1
This post received
KUDOS
Expert's post
GMATinsight wrote:
Quote:
I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?


Factors are not essentially Integers otherwise the expressions of "HCF (Highest Common Factor) of Fractions" would not have existed.

In order to ensure that x is a perfect square we will have to combine the information of first statement with second statement. I hope this will give you an insight why answer option C is the correct answer.


That's not true for the GMAT. Only positive integers are considered as factors and only integers could have factors on the GMAT.
_________________

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

Manager
Manager
avatar
Joined: 17 Mar 2014
Posts: 51
Followers: 0

Kudos [?]: 17 [0], given: 30

If x is an integer, what is the sum of all distinct positive [#permalink] New post 26 Jul 2014, 01:25
Bunuel wrote:
GMATinsight wrote:
Quote:
I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?


Factors are not essentially Integers otherwise the expressions of "HCF (Highest Common Factor) of Fractions" would not have existed.

In order to ensure that x is a perfect square we will have to combine the information of first statement with second statement. I hope this will give you an insight why answer option C is the correct answer.


That's not true for the GMAT. Only positive integers are considered as factors and only integers could have factors on the GMAT.


Thanks for clearing that, should have tested few more values

Few more values that qualify for B
15*17 = 3.K ( for some positive integer K)
here x= 16 and \sqrt {16} = 4 sum of factors =7

63.65= 3.K here x = 64, \sqrt {64} = 8

99.101= 3.k here X= 100, \sqrt {100} =10
etc .

hence B is insufficient

1+2
Only x= 4 qualifies
Expert Post
Veritas Prep GMAT Instructor
User avatar
Joined: 16 Oct 2010
Posts: 4691
Location: Pune, India
Followers: 1089

Kudos [?]: 4882 [0], given: 163

If x is an integer, what is the sum of all distinct positive [#permalink] New post 27 Jul 2014, 20:26
Expert's post
qlx wrote:
I thought statement 2 was sufficient as \sqrt{x} is supposed to be an integer as only integer can have factors .
Based on this, statement 2 says product of 2 consecutive odd integers is odd
(x-1)(x+1)= odd

1*3= 3 here x= 2 but \sqrt{2} is not an integer hence we cannot take x= 2

3*5 = 15 here x= 4 and \sqrt{4} is an integer hence we can take x= 4

5*7 = cannot take, as this is not in the form 3K

7*9=63 here x = 8 but \sqrt{8} is not an integer hence we cannot take x= 8

9*11=99 X=10 but \sqrt{10} is not an integer hence we cannot take x= 10

so from statement 2 after testing various numbers I felt that x = 4 was the only value that qualified hence B was the answer.
Are there any values that I missed ?
or \sqrt{x} need not be an integer?


It is very hard to prove something by just testing numbers. You may not have tested the right numbers or you may not have tested enough numbers. To prove something, use of logic is required. To disprove, it is much easier to use numbers since you only have to find one example where it doesn't hold.

From statement 2, you have [fraction]x^2 = 3x + 1[/fraction]
You need an even perfect square which is 1 more than a multiple of 3. But since \sqrt{x} must be an integer too, we are looking for an even fourth power which is 1 more than a multiple of 3.

An even fourth power is 16 ( = 2^4) which is 1 more than 15. \sqrt{x} = 2 here.
Another even fourth power is 256 ( = 2^8) which is 1 more than 255 (multiple of 3). \sqrt{x} = 4 here.
Another even fourth power is 4096 = 2^{12} which is 1 more than 4095 (multiple of 3). \sqrt{x} = 8 here.

and so on...
_________________

Karishma
Veritas Prep | GMAT Instructor
My Blog

Save $100 on Veritas Prep GMAT Courses And Admissions Consulting
Enroll now. Pay later. Take advantage of Veritas Prep's flexible payment plan options.

Veritas Prep Reviews

If x is an integer, what is the sum of all distinct positive   [#permalink] 27 Jul 2014, 20:26
    Similar topics Author Replies Last post
Similar
Topics:
3 Experts publish their posts in the topic If x and y are distinct positive integers, what is the value ankitmania 4 18 Jul 2010, 07:11
The sum of positive integers x and y is 77. What is the mbunny 7 03 Sep 2007, 17:09
What is the sum of positive integers x and y? (1) sharadGmat 5 27 Jun 2006, 13:14
What is the positive integer n ? (1) The sum of all of the giddi77 13 18 Apr 2006, 23:58
The sum of positive integers x and y is 77. what is the trickygmat 9 17 Oct 2005, 13:10
Display posts from previous: Sort by

If x is an integer, what is the sum of all distinct positive

  Question banks Downloads My Bookmarks Reviews Important topics  


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®.