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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.