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

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