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A and B must be even. (Cause if A is odd and B is even, you will not get an integer.) So, go through the answer choices.

1) Even/2 is even 4/2= 2
2) Even/2 is even again
4) A is already even, so not the answer
5) B is already even, so not the answer.

Which leaves us to C. Remember, when playing odd and even, choosing numbers doesn't always work. If you pick A=10, B=14; then (10+14)/2= 12. And then you're confused.

Look on the number line... 10 . 11 . 12 . 13. 14, the middle number if 12.

But if you choose A=2, B=4.
Look on the number line.. 2 . 3 . 4.. the middle number if 3;
so (2+4)/2 = 3 which is odd.

Remember how to choose numbers in PS and DS will definately help you. Don't choose random numbers, know what you are trying to prove.

I think this problem should be addressed in more structured way:

a/b = even => a = even (it does not matter whether b is odd or even, a will always be even)

a-b = even => There are only two possibilities:
a= odd & b = odd OR a= even & b = even

Considering both statements, we can conclude that a and b both should be even.

Now pick numbers, but while picking the numbers, also check that it satisfy the original statements given in the question. For example, we can not select a = 10 and b = 14. Because although both of them are even, they do not satisfy the first statment a/b = even.

a = 2, 4, 6, ,8, 10, 12.......
b = 2, 4, 6, 8, 10, 12.......

Now find few possible pairs of a and b. (2,2) is not possible (does not satisfy the given equation)

possible pairs could be (4,2), (8, 2), (8,4), (12,2), (12,6)

Now check each answer choice. Only choice 3 can become ODD and that too NOT ALWAYS. Because if a = 8, b = 4, then a+b/2 = 6(even).

So I think the question should probably be worded as "which of the following COULD BE ODD". If it is "which of the following MUST BE ODD" then I guess none of the choices meet the criteria.

I did not skip your post. I was just trying to address the problem in more procedural way. I understand whet you tried to say in your post. And I think I missed the point you were trying to make (in hurry..). This resulted in long and unnecessary method.

I think that the question is missing additional information.

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