$a-b$ even --> either both even or both odd

$\frac{a}{b}$ even --> either both even or $a$ is even and $b$ is odd.

As both statements are true --> $a$ and $b$ must be even.

As $\frac{a}{b}$ is an even integer --> $a$ must be multiple of 4.


Options A is always even.

Options B can be even or odd.

Options C can be even or odd.

Options D: $\frac{a+2}{2}=\frac{a}{2}+1$, as $a$ is multiple of $4$, $\frac{a}{2}$ is even integer --> even+1=odd. Hence option D is always odd.

Options E can be even, odd.


Answer: D.