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

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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.

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

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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
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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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.
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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

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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

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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.
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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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.
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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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
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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Hi shmba,

Good catch! I've updated my explanation.

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

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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 ]

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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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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?
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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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__
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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di  [#permalink]

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

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