GMAT Question of the Day: Daily via email | Daily via Instagram New to GMAT Club? Watch this Video

 It is currently 12 Jul 2020, 22:23

### 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

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

# What is the value of the positive integer n? (1) n^2 + 2n has four di

Author Message
TAGS:

### Hide Tags

Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1339
Location: Malaysia
What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

27 Feb 2017, 23:26
1
25
00:00

Difficulty:

95% (hard)

Question Stats:

46% (02:25) correct 54% (02:32) wrong based on 241 sessions

### HideShow timer Statistics

What is the value of the positive integer n?

(1) $$n^2 + 2n$$ has four distinct positive factors.

(2) $$n^2 + 6n + 8$$ has four distinct positive factors.

_________________
"Be challenged at EVERY MOMENT."

“Strength doesn’t come from what you can do. It comes from overcoming the things you once thought you couldn’t.”

"Each stage of the journey is crucial to attaining new heights of knowledge."

Senior Manager
Joined: 13 Oct 2016
Posts: 352
GPA: 3.98
What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

28 Feb 2017, 05:59
4
3
ziyuen wrote:
What is the value of the positive integer n?

(1) $$n^2 + 2n$$ has four distinct positive factors.

(2) $$n^2 + 6n + 8$$ has four distinct positive factors.

Hi

We'll get 4 distinct factors only in two cases: 1*4 ($$p^3$$) or 2*2 ($$p*q$$), where p and q prime numbers.

(1) $$n^2 + 2n = n(n + 2)$$ has 4 distinct factors, that means we have p*q and our n and n+2 should be prime numbers, in other words we should be able to generate two primes which are in AP with common difference 2.

n=3 (3*5), n=5 (5*7), ... (17*19), (29*31), (41*43) .. Insufficient.

(2) $$n^2 + 6n + 8 = (n + 2)(n + 4$$). As in previous case two primes with distance 2.

n=3 (5*7), n=15 (17*19), n=27 (29*31) ... Insufficient.

(1)&(2) We'll have: n, (n + 2) and (n + 4) are equidistant primes (primes in AP), which leaves us only one choice:

n=3 ---> 3, 5 and 7. There are no more triplets of primes in AP with common difference 2 after the 7. Sufficient.

##### General Discussion
Intern
Joined: 11 Jun 2016
Posts: 4
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

16 Mar 2017, 08:34
Good Question!
Although I don't completely agree with the solution provided above. I think the answer should be E.

Any non-prime interger that is not a perfect square will have an even number of factors, i.e. a certain number of prime factors, the number itself and of course 1, which is a factor of every integer.

Statement 1 tells us that $$n^2$$+2n has 4 distinct factors. This woud also include the number N plus 1. Hence, there are definitely 2 prime numbers as its factor. However, if we pick numbers, the expression seems to work for most numbers. For example, if n=2, then the expression equals 8, which has 4 factors. This works for 3,4,5 ,etc. Insufficient due to no clear result.
hence, n and n+2 are basically 2 prime factors.. We do not know which ones.

Statement 2 gives us the same information with a few more factors. I don't think we can factorize this as that would essentially mean treating the expression as a quadratic equatic, which is incorrect. Again by picking numbers, the expression seems to work for most numbers. For the same reason as statement 1, this statement is also sufficient.

Combining both statements also does not throw out any distinct intger for N. Hence E.

Also, the last bit of the previous explanation by vitaliy doesn't make sense to me. A few primes follow a pattern: they are equidistant by 2 units,i.e. 3,5,7,9,11,13. Hence, the N could be 3 or 5 or 7 or even 9. The pattern does not end after 7.
Senior Manager
Joined: 24 Apr 2016
Posts: 305
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

17 Mar 2017, 11:59
1
What is the value of the positive integer n?

(1) n^2+2n has four distinct positive factors.
This can be written as n * (n+2).
If N is even positive integer 2, then we get 2 * 4 = 8 = 2^3. Total number of distinct positive factors equals 4.
If N is Odd positive integer 3, then we get 3 * 5. Total number of distinct positive factors equals 4.
If N is Odd positive integer 5, then we get 5 * 7. Total number of distinct positive factors equals 4.

As we see that there are multiple values of N which satisfy the above, Statement 1 is insufficient.

(2) n^2+6n+8 has four distinct positive factors.
This can be written as (n+2)(n+4)

If N is even positive integer 2, then we get 4 * 6 = 24 = (2^3) * 3. Total number of distinct positive factors equals 8. So this is out
If N is Odd positive integer 1, then we get 3 * 5. Total number of distinct positive factors equals 4.
If N is Odd positive integer 3, then we get 5 * 7. Total number of distinct positive factors equals 4.

As we see that there are multiple values of N which satisfy the above, Statement 2 is insufficient.

Combining both statements we see that N=3, satisfies both. Hence answer is C.
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 17071
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

Updated on: 29 Aug 2018, 19:58
1
1
Hi All,

We're told that N is a positive integer. We're asked for the value of N. This question can be solved by TESTing VALUES and a rare Number Property rule: for a number to have exactly 4 factors, that number must be the product of two different prime numbers OR the cube of a prime number.

For example:
(2)(3) = 6 and its factors are 1, 2, 3 and 6
(2)(2)(2) = 8 and its factors are 1, 2, 4 and 8

With the given information in Facts 1 and 2, we can ‘rewrite’ the expressions as a product of 2 values to see how many different ‘pairs’ of prime numbers are possible.

1) N^2 + 2N has 4 distinct positive factors.

N^2 + 2N can be rewritten as N(N+2), so what COULD N be so that BOTH N and (N+2) are prime numbers or a cubed prime….

N could be 2, meaning the product would be (2)(4) = (2)(2)(2)
N could be 3, meaning the product would be (3)(5)
N could be 5, meaning the product would be (5)(7)
N could be 11, meaning the product would be (11)(13)
Etc.
Fact 1 is INSUFFICIENT

2) N^2 + 6N + 8 has 4 distinct positive factors.

N^2 + 6N + 8 can be rewritten as (N+2)(N+4). In the same way that we handled Fact 1, what COULD N be so that BOTH (N+2) and (N+4) are prime numbers or a cubed prime….

N could be 1, meaning the product would be (3)(5)
N could be 3, meaning the product would be (5)(7)
N could be 9, meaning the product would be (11)(13)
Etc.
Fact 2 is INSUFFICIENT

Combined, we can’t use N=2 (since it does not ‘fit’ Fact 2) and N, (N+2) and (N+4) would ALL have to be primes. Put another way, we need 3 CONSECUTIVE ODD integers that are ALL prime. That will only occur when N = 3… meaning the three numbers would be 3, 5 and 7. In any other circumstance, we will end up with at least one non-prime odd number among the 3 integers.
Combined, SUFFICIENT

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com

The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★

Originally posted by EMPOWERgmatRichC on 09 Dec 2017, 15:47.
Last edited by EMPOWERgmatRichC on 29 Aug 2018, 19:58, edited 1 time in total.
Current Student
Joined: 16 Aug 2018
Posts: 27
Concentration: General Management, Strategy
Schools: Guanghua"21 (A)
GMAT 1: 700 Q49 V36
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

27 Aug 2018, 19:51
1
EMPOWERgmatRichC wrote:
Hi All,

We're told that N is a positive integer. We're asked for the value of N. This question can be solved by TESTing VALUES.

1) N^2 + 2N has 4 distinct positive factors.

IF....
N=3, then N^2+2N = 15 (factors are 1, 3, 5 and 15)
N=5, then N^2+2N = 35 (factors are 1, 5, 7 and 35)
Fact 1 is INSUFFICIENT

2) N^2 + 6N + 8 has 4 distinct positive factors.

IF....
N=3, then N^2+6N+8 = 35 (factors are 1, 5, 7 and 35)
N=5, then N^2+6N+8 = 63 (factors are 1, 7, 9 and 63)
Fact 2 is INSUFFICIENT

Combined, we already have two different values for N that 'fit' both Facts.
Combined, INSUFFICIENT

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

Hi Rich,
I don't agree with your explanation on statement2.
if N=5,(N+2)(N+4)=5*9=5*3^2. it has (1+1)*(2+1)=6 factors. not four.
3 seems to be the only value that makes N,N+2,N+4 all primes.
RSM Erasmus Moderator
Joined: 26 Mar 2013
Posts: 2474
Concentration: Operations, Strategy
Schools: Erasmus
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

29 Aug 2018, 18:56
EMPOWERgmatRichC wrote:
Hi All,

We're told that N is a positive integer. We're asked for the value of N. This question can be solved by TESTing VALUES.

1) N^2 + 2N has 4 distinct positive factors.

IF....
N=3, then N^2+2N = 15 (factors are 1, 3, 5 and 15)
N=5, then N^2+2N = 35 (factors are 1, 5, 7 and 35)
Fact 1 is INSUFFICIENT

2) N^2 + 6N + 8 has 4 distinct positive factors.

IF....
N=3, then N^2+6N+8 = 35 (factors are 1, 5, 7 and 35)
N=5, then N^2+6N+8 = 63 (factors are 1, 7, 9 and 63)
Fact 2 is INSUFFICIENT

Combined, we already have two different values for N that 'fit' both Facts.
Combined, INSUFFICIENT

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

Dear EMPOWERgmatRichC

In statement 2, if N=5, then 63 will have 6 factors (1,3,7,9, 21, 63)......So 5 is invalid.

Thanks
EMPOWERgmat Instructor
Status: GMAT Assassin/Co-Founder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 17071
Location: United States (CA)
GMAT 1: 800 Q51 V49
GRE 1: Q170 V170
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

29 Aug 2018, 20:00
Hi shmba,

Good catch! I've updated my explanation.

GMAT assassins aren't born, they're made,
Rich
_________________
Contact Rich at: Rich.C@empowergmat.com

The Course Used By GMAT Club Moderators To Earn 750+

souvik101990 Score: 760 Q50 V42 ★★★★★
ENGRTOMBA2018 Score: 750 Q49 V44 ★★★★★
SVP
Joined: 23 Feb 2015
Posts: 1946
What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

06 Feb 2019, 08:58
1
hazelnut wrote:
What is the value of the positive integer n?

(1) $$n^2 + 2n$$ has four distinct positive factors.

(2) $$n^2 + 6n + 8$$ has four distinct positive factors.

Hi Bunuel,
Someone may be misguided by the word "Highlight" in the "Rules of posting", because we get a shortcut button namely "highlight" when make new topic. Please see the attachment.

https://gmatclub.com/forum/rules-for-po ... l#p1096628
Attachments

nnnnnnnn.PNG [ 14.77 KiB | Viewed 1026 times ]

posting rules.PNG [ 70.23 KiB | Viewed 1029 times ]

Math Expert
Joined: 02 Sep 2009
Posts: 65194
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

06 Feb 2019, 10:32
hazelnut wrote:
What is the value of the positive integer n?

(1) $$n^2 + 2n$$ has four distinct positive factors.

(2) $$n^2 + 6n + 8$$ has four distinct positive factors.

Hi Bunuel,
Someone may be misguided by the word "Highlight" in the "Rules of posting", because we get a shortcut button namely "highlight" when make new topic. Please see the attachment.

https://gmatclub.com/forum/rules-for-po ... l#p1096628

I see what you mean. Would Mark be better? Or?
_________________
SVP
Joined: 23 Feb 2015
Posts: 1946
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

06 Feb 2019, 10:41
1
Bunuel wrote:
hazelnut wrote:
What is the value of the positive integer n?

(1) $$n^2 + 2n$$ has four distinct positive factors.

(2) $$n^2 + 6n + 8$$ has four distinct positive factors.

Hi Bunuel,
Someone may be misguided by the word "Highlight" in the "Rules of posting", because we get a shortcut button namely "highlight" when make new topic. Please see the attachment.

https://gmatclub.com/forum/rules-for-po ... l#p1096628

I see what you mean. Would Mark be better? Or?

Actually, I'm a non-native speaker, so it will be wrong decision to make any suggestion (for me) for this specific word. The word "mark" is perfectly fine at least to me. So, if the word "mark" also makes sense to native and non-native then the problematic word (highlight) should be replaced with "mark".
Thanks__
Non-Human User
Joined: 09 Sep 2013
Posts: 15425
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

### Show Tags

01 Jun 2020, 20:36
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.
_________________
Re: What is the value of the positive integer n? (1) n^2 + 2n has four di   [#permalink] 01 Jun 2020, 20:36