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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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28 Feb 2017, 05:59

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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.

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

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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. 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.

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

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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.