$D := (-1)^{\frac{m-1}{2}}n$ and $N := (-1)^{\frac{n-1}{2}}n$. Show that $N = D$.

Question

Let $m,n$ be an odd, positive integer:

$D := (-1)^{\frac{m-1}{2}}n$ with D $\equiv 1$ (mod 4)

$N := (-1)^{\frac{n-1}{2}}n$ with N $\equiv 1$ (mod 4)

Show that $N = D$:

What I got so far:

$D = \pm n$ and it has to be odd since $n$ is odd. Same for $N$. Why can I follow from that that $D = N$?

Answer 1

Hint: From the assumption each of $N,D$ is either $n$ or $-n$.

If $n \equiv 1 \text{ mod } 4$ then $-n \equiv 3 \text{ mod } 4$ and vice versa.

Is it possible that $N=n$ and $D=-n$?