Vincen wrote:If positive integers a and b are both odd, which of the following must not be odd?

A) ab
B) a(ab)
C) b - a - 1
D) (a + b)/2
E) a - b

The OA is E.

If we set a=3 and b=5 the (a+b)/2 is 4 and it is not odd. Why option D is wrong?

Given a and b are both odd. Just put a = 1 and b = 1, then check all the options:
(A) ab = 1*1 = 1 Odd
(B) a*(ab) = 1*(1*1) = 1 Odd
(C) b - a - 1 = 1 - 1 - 1 = -1 Odd
(D) (a+b)/2 = (1+1)/2 = 1 Odd
(E) a - b = 1 - 1 = 0 Even

So, the answer is E

You asked about D specifically, so I'll only address that.

It actually comes down to the remainder of our odd integers when divided by 4. If they have different remainders (one has remainder 1, one has remainder 3), then the average of the two will be even. If they have the same remainder by 4 (both remainder 1 or both remainder 3), then the average will be odd.

A few examples:

5 + 7 => 12, average = 6
5 + 9 => 14, average = 7
7 + 11 => 11, average = 9

As a good follow up exercise, see if you can explain why this must be. (Hint: a multiple of 4 can be written as 4*some integer.)