# How do you solve a system of equations that involves modular arithmetic?

Say I wanted to determine all the times when the minute hand and the hour hand point at the exact same number. I decide to measure the hands' positions by how far they have rotated from directly vertical (e.g., 12:00$=0$, 3:00$=\pi/2$, 6:00$=\pi$). I write these positions as a function of $t$, which is measured in minutes and ranges between $0$ and $720$ since the pattern repeats every $12$ hours and there are $720$ minutes in one hour.

Here are my equations (I got them off of the AP Physics C formula chart):

$T=2\pi/\omega$

$T_m=60\Longrightarrow\omega_m=\pi/30$

$T_h=720\Longrightarrow\omega_h=\pi/360$

Therefore,

$\theta_m=\pi t/30$

$\theta_h=\pi t/360$

Now, the catch is that the angle measure repeats every $2\pi$, so my final system of equations is

$$\left\{ \begin{array}& \theta_m=\frac{\pi t}{30}\mod 2\pi \\ \theta_h=\frac{\pi t}{360}\mod 2\pi \\ \end{array} \right.$$

And I want to solve for $$\left\{ t\in\left[ 0,720\right) : \theta_m=\theta_h\right\}$$