Clock Squares

Why do this problem?

This problem follows on from Clock Arithmetic. It gives students the opportunity to consider what happens when we square numbers in modular arithmetic.  It gives students the opportunity to work systematically and to spot patterns.

Possible approach

This problem featured in an NRICH Secondary webinar in April 2021.

To start with students could be asked what the values of $1^2$, $2^2$, $3^2$ and $4^2$ are modulo $5$.  Can they state what $5^2 \text{ mod } 5$ is without evaluating $5^2$?  Can they predict what $6^2, 7^2$ etc are modulo $5$?  

Students could work in pairs to find square numbers in different modulos, and it might be useful to start by only considering prime number modulos.  Encourage students to collect their results in a table and to start to look for patterns.  Challenge them to predict what the pattern will look like for a new modulo and test their predictions.

Students can then go on to investigate square numbers in modulos that are odd but not prime, or which are even.

Key questions

Which values of $x$ satisfy $x^2 \equiv 0 \text{ mod }n$?

When working in modulo $p$, where $p$ is prime, how many different values of $x^2 \text{ mod } p$ are there?

Possible support

Students could start by working on Clock Arithmetic.
Students should be encouraged to work systematically and to lay out their results logically, possibly in a table.

Possible extension

If they have not already done so, students might like to move onto More Adventures with Modular Arithmetic.  They might also like to investigate Euler's Totient Function.

Further reading on modular arithmetic can be found here