Is the positive integer x even?

From the question stem, we get that:
\(x\) is an integer and \(x>0\)
Now lets consider the statements:

Statement 1
\((x - 1)\) is a prime number
This gives us 2 scenarios:
either \((x - 1)\) is an even prime number (i.e. 2) or \((x - 1)\) is an odd primer number (i.e. any prime numbers except 2)
If \((x - 1)\) is an even prime number, then \(x = 3\), an odd integer
If \((x - 1)\) is an odd prime number, then \(x\) has to be an even integer
Therefore, we do not get a definitive yes/no from this statement. This statement is insufficient.
Hence, we can reject options A and D.

Statement 2
\((x^2 - 1)\) is a prime number.
This also gives us 2 scenarios:
either \((x^2 - 1)\) is an even prime number (i.e. 2) or \((x^2 - 1)\) is an odd primer number (i.e. any prime numbers except 2)
If \((x^2 - 1) = 2\); \(x^2 = 3\). This is not possible as we know that \(x\) is an integer.
Hence, the only possible scenario is that \((x^2 - 1)\) is an odd primer number.
This implies \(x^2\) is an even number and accordingly, \(x\) is an even number.
This statement is sufficient. We can reject options C and E.

The correct answer is B.

Excellent Question
Here in statement one prime number can be 2 so X can be even or odd
But in statement two x^2-1 can never be two as it would mean that x is not an integer which is contradictory
so x^2-1 must be odd prime => x^2= odd+odd=even so x must be even
Smash that B

This is how i solved

Statement 1: (x - 1) is a prime number
x = 3 implies (x - 1) = 2. A prime number
x = 4 implies (x - 1) = 3. A prime number.
Hence x can be both even and odd. Insufficient

Statement 2: (x^2 - 1) is a prime number
Only possible value of X^2 -1 being a prime no is 2^2 -1 = 3 Which is not even but an odd prime..

rest when u test values we dont get any value that is prime

Eg : 11^2-1 = 121-1 = 120 Not prime
13^2-1 = 198-1 = 168 not prime

3^2- 1 =8 Not prime ... So only prime no that satisfies the equation is 2^2-1 = 3 Which is an odd prime and we can answer the question with certainty

Alternatively
(x - 1)(x + 1) is a prime number.
The only possible value of (x - 1)(x + 1) can be 1*3 = 3
Therefore x = 2
Sufficient
Option B

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.