First, the question really asks: "Is k prime?"

From 1)- knowing k is larger than 4! is not enough. There is an infinite number of primes; this kind of statement cannot possibly help.

From 2)- k is between 13!+2 and 13!+13. None of the numbers in this list can be prime. I'll take one example, 13! + 11. Notice that 13! is a multiple of 11 (it's 13*12*11*...). When you add two multiples of 11, you get another multiple of 11. 13!+11 is a multiple of 11, and is not prime. This statement is sufficient.

In full detail:

13! + 11 = 11*(13*12*10! + 1)
13! + 11 is thus a multiple of 11 and also a multiple of 13*12*10! + 1, and is not prime.