The following is from Hardy's An Introduction to the Theory of Numbers:
A lecture is given on every alternate day (including Sundays), and that the first lecture occurs on a Monday. When will a lecture first fall on a Tuesday? If this lecture is the $(x + 1 )$th then $2x\equiv 1\pmod{7}$
One can find by trial that the least positive solution is $x=4$. Thus the fifth lecture will fall on a Tuesday and this will be the first that will do so.
It is not hard to solve the equation. However, I don't understand how to actually build up this equation. I guess $2$ is from the "alternate day" and $7$ is from the number of the days in a week. $1$ may represent "Monday".(or "Sunday"?) But what does $2x$ mean and why does $(x+1)$ become $x$ and $2x\equiv 1\pmod{7}$, i.e., $7|(2x-1)$?