nt.number theory – Orbits of $mathbb{Z}/nmathbb{Z}$ under the action of its multiplicative group $left(mathbb{Z}/nmathbb{Z}right)^{times}$

Let $mathbb{Z}/nmathbb{Z}$, $left(mathbb{Z}/nmathbb{Z}right)^{times}$, and $mathcal{D}_n$ be the ring of integers modulo $n$, its multiplicative group of order $n$, and the set of divisors of $n$, respectively. The map $phi: left(mathbb{Z}/nmathbb{Z}right)^{times}timesmathbb{Z}/nmathbb{Z}rightarrowmathbb{Z}/nmathbb{Z}$, given by $phi(g,x):=gx$ is a left group action on $mathbb{Z}/nmathbb{Z}$. I am having a hard time in showing that $phi$ decomposes $mathbb{Z}/nmathbb{Z}$ into $|mathcal{D}_n|$-many distinct orbits given by $left(mathbb{Z}/nmathbb{Z}right)^{times}cdotbar{delta}$, where $delta$ is a divisor of $n$. Let me give an example. If $n=12$, then $mathbb{Z}/12mathbb{Z}={bar{0},dots,bar{11}}$, $left(mathbb{Z}/12mathbb{Z}right)^{times}={bar{1}, bar{5},bar{7}, bar{11}}$, and $mathcal{D}_{12}={1,2,3,4,6,12}$ and we have:
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{1}={bar{1}, bar{5},bar{7}, bar{11}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{2}={bar{2}, bar{10}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{3}={bar{3}, bar{9}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{4}={bar{4}, bar{8}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{6}={bar{6}},$$
$$left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{0}={bar{0}},$$
and
$$bigcup_{delta|12}left(mathbb{Z}/12mathbb{Z}right)^{times}cdotbar{delta}=mathbb{Z}/12mathbb{Z}.$$
I would be very grateful for any help or insights.