November 22, 2018 November 22, 2018 10:00 PM PST 11:00 PM PST Mark your calendars  All GMAT Club Tests are free and open November 22nd to celebrate Thanksgiving Day! Access will be available from 0:01 AM to 11:59 PM, Pacific Time (USA) November 23, 2018 November 23, 2018 10:00 PM PST 11:00 PM PST Practice the one most important Quant section  Integer properties, and rapidly improve your skills.
Author 
Message 
TAGS:

Hide Tags

Senior SC Moderator
Joined: 14 Nov 2016
Posts: 1317
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
Question Stats:
45% (02:24) correct 55% (02:27) wrong based on 205 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.
Official Answer and Stats are available only to registered users. Register/ Login.
_________________
"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."
Rules for posting in verbal forum  Please DO NOT post short answer in your post!
Advanced Search : https://gmatclub.com/forum/advancedsearch/




Senior Manager
Joined: 13 Oct 2016
Posts: 367
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
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. Answer C.




Intern
Joined: 11 Jun 2016
Posts: 5

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 nonprime 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. Please help me understand how C is correct, instead of 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: 331

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
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/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12891
Location: United States (CA)

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
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 nonprime odd number among the 3 integers. Combined, SUFFICIENT Final Answer: 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
CoFounder & 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!*****



Intern
Joined: 16 Aug 2018
Posts: 29
Concentration: General Management, Strategy

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
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 Final Answer: 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.



SVP
Joined: 26 Mar 2013
Posts: 1887

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 Final Answer: GMAT assassins aren't born, they're made, Rich Dear EMPOWERgmatRichCIn 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/CoFounder
Affiliations: EMPOWERgmat
Joined: 19 Dec 2014
Posts: 12891
Location: United States (CA)

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
_________________
760+: Learn What GMAT Assassins Do to Score at the Highest Levels Contact Rich at: Rich.C@empowergmat.com
Rich Cohen
CoFounder & 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!*****




Re: What is the value of the positive integer n? (1) n^2 + 2n has four di &nbs
[#permalink]
29 Aug 2018, 20:00






