Amit is right about this process of examining statement 2. I just want to take one step back and look at the question itself b/c there's a very easy way to rephrase it. This is basically asking us whether k is prime.

A prime number has exactly two factors: itself and 1. By definition, it cannot have a factor p that is between itself and 1. A non-prime number, on the other hand, that is greater than 1 will have a factor between itself and 1 (again, by definition).

So I can answer this question if I know whether k is prime.

(1) just tells me that k>24. There are prime numbers and non-prime numbers greater than 24, so that's not useful.

(2) for any sum, if the two numbers in that sum have a common factor, that factor will also be a factor of the sum. E.g., 2 + 4 = 6. 2 is a factor of 2 and 2 is a factor of 4. Therefore, 2 will also be a factor of 6.

Now, to follow amit's logic:
13! + 2 = 13*12*11*10*9*8*7*6*5*4*3*2 + 2. 2 is a factor of both of the numbers, so the sum will also have 2 as a factor. Therefore, the first possibility is not prime. Follow this logic all the way through 13! + 13; you will always have at least one factor, so that number is not prime. All of the possibilities given by statement 2 are not prime so I can answer the question: No, the integer k does not have a factor p such that 1<p<k.