If a, b, and c are consecutive odd positive integers and a

Actually this question does not test any obscure knowledge.

The moment you read statement I, you should know that this is not necessarily true. You don't have to test any numbers. This is because you should know a very basic thing about prime numbers - you cannot find a pattern to them i.e. they appear randomly. You cannot say when exactly the next prime number will appear. If it were necessary that out of three consecutive odd numbers, one needs to be prime then there is a pattern to finding them. Hence (I) has to be false.

II. ab>c
This seems would be true often since 5*7 will be greater than 9 etc but what if a is a very small number. So then let's say a = 1.
1*3 < 5
So (II) is not necessarily true.

Then looking at the options III must be true and answer must be (A).

If you want to confirm,
III. a + b + c = 3b
(2X - 1) + (2X + 1) + (2X + 3) = 6X + 3 = 3*(2X + 1)