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

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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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New post 28 Feb 2017, 00:26
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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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New post 28 Feb 2017, 06:59
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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.

Answer C.
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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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New post 16 Mar 2017, 09:34
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.
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.
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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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New post 17 Mar 2017, 12:59
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.
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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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New post 09 Dec 2017, 16:47
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

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Re: What is the value of the positive integer n? (1) n^2 + 2n has four di &nbs [#permalink] 09 Dec 2017, 16:47
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