Days distinction between mars and earth orbiting Assuming that Earth orbits the Sun over a period of $365$ Earth days, and Mars orbits the Sun over a period of $687$ Earth days. The earth orbit starts at Day $0$ and continues to Day $364$ and starts over at Day $0$. Mars orbit is similar but is on a $687$-day time-scale instead. How can I determine how long it will take until both planets are on day $0$ of their orbits at the same time? Let me give you an example: If Earth is on Day $364$ and if Mars is on Day $686$, the smallest number of days till the two planets will both be on day $0$ of their orbit is $1$. However, if Earth is at Day $0$ and Mars at Day $1$, the smallest number of days till the two planets will both be on day $0$ of their orbit is $239075$.

You want to solve $365k\equiv 1 \pmod {687}$, which is solved by $k\equiv 32 \pmod {687}$. This means $365\cdot 32=11680$ is what you seek.