Data Sufficiency

Please help.
The OA is B and I don't know why.
Marc,
Thanks.

Thanks Rich,

It makes more sense to me now.

Marc.

They're annoying, but literal definitions are really useful on the GMAT. If we remember the literal definition of a prime ("a positive integer that has exactly two factors: 1, and itself"), we can see that S2 simply states this definition, forcing p to be prime.

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If integer p is greater than 1, is p a prime number?
1) p is odd
2) The only positive factors of p are 1 and p

There is one variable (p) and 2 equations are given from the 2 conditions, making (D) our likely answer.
From condition 1, the answer is 'yes' for p=3, but 'no' for p=9, so this is insufficient.
For condition 2, it is sufficient as it is the definition for a prime number. The answer is (B).

For cases where we need 1 more equation, such as original conditions with "1 variable", or "2 variables and 1 equation", or "3 variables and 2 equations", we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.