varun289 wrote:278. If x is an integer, does x have a factor n such that 1 < n < x?
(1) x > 3!
(2) 15! + 2 ≤ x ≤ 15! + 15

This is a great candidate for rephrasing the target question.
What kinds of integers have a factor n such that 1 < n < x? Non-prime integers. So, we're really just asking whether or not n is a non-prime integer. Let's make it even easier on ourselves and ask . . .

Rephrased target question: Is x prime?

Statement 1: x > 3!
In other words, x > 6
case a) x = 7, in which case x is prime
case b) x = 8, in which case x is not prime
Statement 1 is NOT SUFFICIENT

Statement 2: 15! + 2 ≤ x ≤ 15! + 15
This is saying that x can have one of 14 different possible values. So, let's begin checking some values.

Is 15! + 2 prime? No.
Notice that 15! = (15)(14)(13)...(3)(2)(1)
So, we can factor a 2 out of 15! + 2, to get:
15! + 2 = 2[(15)(14)(13)...(3)(1) + 1]
This means that 2 is a factor of 15! + 2, which means it is not prime.

Next, 15! + 3 prime? No.
Notice that 15! = (15)(14)(13)...(4)(3)(2)(1)
So, we can factor a 3 out of 15! + 3, to get:
15! + 3 = 3[(15)(14)(13)...(4)(2)(1) + 1]
This means that 3 is a factor of 15! + 3, which means it is not prime.

We can continue this process to show that none of the 14 possible values of x are prime.
As such, statement 2 is SUFFICIENT and the answer is B

Cheers,
Brent