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

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.

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

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.

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

Hi shmba,

Good catch! I've updated my explanation.

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

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?

Thanks for your kudos.
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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