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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., $7|(2x-1)$?

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(x+1) is to accommodate our human way of counting starting at 1. – starblue Jun 17 '11 at 21:05
up vote 7 down vote accepted

Every lecture occurs $2$ days later than the previous one. You can count how many days have passed since Monday then by $2x$. The point is after jumping forward two days $x$ times, you want to ask yourself when will you be only $1$ day later in the week?, since Tuesday is one day after Monday. This is why you want to solve $2x\equiv 1\pmod{7}$. Since a week has $7$ days, you take the congruence modulo $7$.

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