Does it matter if the scaling term is in the DFT versus the inverse DFT?
It's mostly a matter of convention.
There are three options. $1/N$ for the inverse transform, $1/N$ for the forward transform and $1/\sqrt{N}$ for both. The first is the one most commonly used, the last one has the advantage that Parseval's theorem holds directly, i.e. $$\sum |x_n|^2 = \sum |X_k|^2$$