If $\displaystyle n $ is a natural number, then

$\displaystyle (2003 + \frac{1}{2})^n + (2004 + \frac{1}{2})^n $ is a positive integer:

A) when $\displaystyle n $ is even

B) when $\displaystyle n $ is odd

C) only when $\displaystyle n=117$ or $\displaystyle n=119$

D) only when $\displaystyle n=1$ or $\displaystyle n=3$

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Simplifying the expression,

$\displaystyle (\frac{4007^n + 4009^n}{2^n})$

Therefore, $\displaystyle (4007^n + 4009^n)$ must be divisible by $\displaystyle (2^n)$

But how do I proceed from here? Can it be done without the binomial theorem?