Interesting example of linear congruence

Question

I am about to give a talk on linear congruence to high school students. Can you suggest an example of linear congruence that will immediately make students fall in love with linear congruence?

Answer 1

I like this application in astronomy:

The three comets 4P/Faye, 8P/Tuttle and 2P/Encke have orbital periods 8, 13 and 3 years. The previous three Perihelions were in 2017, 2014 and 2018. In which year will the comets reach their Perihelions at the same time? (For simplification assume that time is measured in integers and orbital periods are constant).

Answer 2

If you go as far as the Chinese Remainder Theorem, there is the trick of asking a student to think of a number $n$ between $0$ and $59$ (inclusive). Then have the student give the remainders $r_3$, $r_4$, $r_5$ of dividing $n$ by $3, 4$, and $5$ respectively.

You calculate $40r_3+45r_4+36r_5$, and reduce the total $\pmod{60}$. This should give you back the student's number. With a bit of practice, you can do all the calculations mentally.