Solving linear equation modulo two pi I have an equation I'd like to solve given as $$s \cdot \alpha \equiv p  (\mathrm{mod} 2 \pi) $$ The numbers $s$ and $p$ are known, while $\alpha$ is to be solved for, and additionally is between $0$ and $2 \pi$. I have found a solution using other information, so I know solutions exist, but I'd like to know if solutions are unique and more importantly a general way of calculating them from the given information.

$$\begin{align} s\alpha\, &\equiv\, p\!\!\pmod{\!2\pi \Bbb Z}\\[.3em]
\iff\  s\alpha\, &=\, p\ +\ 2\pi n,\quad {\rm for\ some}\ \ n\in\Bbb Z\\[.3em]
\iff\ \ \  \alpha\, &=\, \dfrac{p}s + \dfrac{2\pi}s\, n,\,\ \ {\rm for\ some}\ \ n\in\Bbb Z\\[.3em]
\iff\ \ \ \alpha\, &\equiv\, \dfrac{p}{s}\!\!\pmod{\!\dfrac{2\pi}s\, \Bbb Z}
\end{align}$$
