Mechmeera wrote:source Kaplan

x is a positive integer and (x − 1) is prime. Is x a prime number?

(1) x + 2 is prime.

(2) x + 1 is not prime.

List options for x-1:
x-1 = 2, 3, 5, 7, 11, 13, 15, 17...

Adding 1 to each value in the list, we get the following options for x:
x = 3, 4, 6, 8, 12, 14, 16, 18...

Notice that x=3 is the only ODD option for x.

Statement 1: x+2 is prime
Of the options for x, only x=3 satisfies the constraint that x+2 is prime:
3+2 = 5.
Since the remaining options for x are even, each will yield an EVEN -- and thus, NONPRIME -- value for x+2.
Thus, x=3.
SUFFICIENT.

Statement 2: x+1 is not prime
It's possible that x=3, since 3+1 = 4, which is not prime.
It's possible that x=5, since 5+1 = 6, which is not prime.
Since x can be different values, INSUFFICIENT.

The correct answer is A.

Another idea is to list the first few primes:

2, 3, 5, 7, 11, 13, 17, ...

Notice that every prime other than 2 is odd. So the distance between ANY two odd primes will be Odd - Odd, which is Even.

S1:: (x - 1) and (x + 2) are both prime

This has a distance of 3, which is NOT even. So we must have 2 and 5. Hence x = 3; SUFFICIENT.

S2:: (x + 1) is not prime

This doesn't help: we could have x = 3 or x = 8.