Need help proving an odd number divided by 2 is never an integer

I know it's true. This is actually part of a bigger proof, but I got it down to this, but I can't just say it's true. I have to somehow show it's true by relating it to an integer.

$\displaystyle \frac{2k+1}{2}$

where 2k+1 is the definition of an odd integer. I can rearrange it and get k + 1/2, but I still need to prove that k + 1/2 is not an integer. I can't just say it. The only thing we have learned about that can identify integers is divisibility.

i.e. a|b means a = bk for some k that's an integer

Anyone know how I can prove this?

If it helps, the original question was:

$\displaystyle \forall q \in \textbf{Q}, \exists r \in\textbf{Q}$ so that q + r is not an integer

Which I figured I could do two cases where the rational number q is an integer, or when it isn't. So now that I'm at the case where it is an integer, I'm trying to prove that adding 1/2 (r value I picked) to it makes it so it's no longer an integer. Which got me to (2k+1)/2