I don't understand how how they came to these equations

Question

Trying to understand a particular equation, and it mentions these inequalities but i don't fully understand the logic of how they solved for n and m:

Hope some one can explain :)

Answer 1

With $$-\frac{N}{2} \leq n < \frac{N}{2}$$

Add $\frac{N}{2}$ to everything and you get $$ 0 \leq n + \frac{N}{2} < N$$

Comparing this with $0 \leq n' < N$, we can set $n' = n + \frac{N}{2}$.