Playing In The Gaussian Plane

Let $F=\mathbb{Z}[i]/(8+3i)$ . We know from here that $F$ has $N(8+3i)=8^2+3^2=73$ elements. Now 73 is prime in $\mathbb{Z}$ , so that $8+3i$ is prime in $\mathbb{Z}[i]$ and $F$ is a field. The multiplicative group of a finite field is cyclic; let $g$ be a generator.

Let's turn to the equation $z^{120}=1$ in $F$ . We can write $z=g^m$ for $1\leq m\leq 72$ , with $g^{72}=1$ , and the equation takes the form $g^{120m}=1$ . The solutions $m$ are of the form $72|120m$ or $3|5m$ or $3|m$ . Thus we have the $\boxed{24}$ solutions $m=3k$ for $k=1...24$ .