Square Remainders

Why do this problem?

This problem asks students to use number theory to prove some results, and also show that some results are not possible.

Possible approach

To begin with students could consider what happens when they divide square numbers by $8$ by working systematically through the first few numbers.

If students are not familiar with standard expressions for odd and even numbers then they could be asked to find the rule for the $n$th term of the sequences:

• $2, 4, 6, 8, 10, ...$ $n$th term is given by $2n$
• $1, 3, 5, 7, 9, ...$ $n$th term is given by $2n-1$, although $2n+1$ might be nicer to use!

Key questions

• If an odd number can be written as $2k+1$, what would be the expression for this number squared?
• If we want to show that adding two odd numbers gives an even number, should we consider $(2a+1)+(2a-1)$ or $(2a+1)+(2b+1)$?  Why?
• What can you say about the parity of the two numbers $k$ and $k+1$?  The parity of a number is referring to whether it is odd or even.
• If you add two numbers of the form $8a+1$ and $8b+1$, what would be the remainder if you divided the sum by $8$?

Possible extension

Here is a list of number theory problems.