Times Tables and Modulo Arithmetic

Great fun, Modulo Arithmetic. It is sometimes called ‘Remainder Arithmetic’ and by the time you have finished this task, you will see why.

Imagine a clock. (Who said mathematics was difficult? Or times tables, come to that?).

As you know, it has twelve numbers on it, starting at 1 and finishing at 12. Pretty obvious really.

Anyway, most clocks have 12 numbers. But…but…but… in Modulo Arithmetic we imagine clocks, but we replace the number at the top with zero.

And…and…and… in Modulo Arithmetic we can have clocks with any number of numbers on them, provided they start at zero.

We call this a Module 6 clock and we can do Modulo 6 arithmetic on it.

Let’s start with addition. You will notice that as we only have the numbers 0 to 5, these are the only numbers we can add, so our addition sum would have to be something like:

                     3 + 2

This really says start at the number 3 and move 2 places clockwise to get the answer.

I am sure you will agree this is 5, so:

                   3 + 2 = 5

Which is simple enough.

Now try:     4 + 5

This means start at 4 and go 5 places around the clock as before, but this takes us past 0 and on to 3, so:

                4 + 5 = 3

I told you this was fun!

I have put in the two answers we have already found. Try to complete the rest of the table.

I expect you can see the patterns in the table immediately.

‘Okay,’ I hear you say, ‘that’s great, fun, simple, better than playing football, but what has it got to do with times tables?’

The answer is that we can multiply as well as add and we can make up a times table just like the addition table (but it’s going to be a bit harder, of course!).

Let’s start with a very easy multiplication sum:

             1 x 3

This means start at zero and add on 1 three times. Do this by counting around the clock and the answer is obviously 3, so:

            1 x 3 = 3

Let’s try a harder one:

             3 x 4

To do this begin at 0 again and count round 3 spaces four times.

This should bring you to 0, so:

        3 x 4 = 0

Now try:

             5 x 3

Start at 0 and count round 5 three times. This should bring you to 3, so

                 5 x 3 = 3

Do you have the idea?

If you do, you should be able to complete the times table below. This is similar to the addition table, but you are going to multiply instead of add.

Before you begin, there is just one thing to be aware of. What happens when we multiply by 0? Well, the answer is always 0 just as it would be with normal numbers.

Think of it this way. If you have the sum  4 x 0, for instance, this means start at 0 as usual and go 4 spaces no times. In other words – don’t move. So you are still at 0.

If you have 0 x 4, this means start at 0 and move 0 places four times, which also leaves you at 0.

Now have a go at completing the table.

To make sure you are doing this correctly, I have put the answer at the end of this page, but don’t look at it until you have tried the whole table yourself.

Once you have mastered this, you are getting on really well, so you are ready to try another modulo.

You will remember that the one we have just used is called Modulo 6. It has the numbers 0 to 5.

Let’s try some multiplications such as:

               7 x 5

Again we begin at 0 and count round the circle 7 spaces five times. This should bring you to 3, so:

            7 x 5 = 3

Let’s try one more: What is the answer to:

            5 x 3

Start at 0 as always and count round 5 spaces three times. This will bring you to 7, so:

            5 x 3 = 7

Next I obviously want you to complete the multiplication table for Modulo 8, but first I can show you a short cut which will make your life easier, especially when the modulo is large.

Let’s take the last sum as an example. In normal arithmetic, 5 x 3 = 15, of course, so altogether we need to count 15 spaces from 0. This will bring us to 7 as we already know.

But every time we count round 8 spaces we get back to 0, so all we have to do is to calculate the remainder when we divide by 8 (as we are using Modulo 8).

15 divided by 8 is 1 remainder 7, so 7 is the answer. Isn't it a good job you know your times tables really well?

Let’s try another one:

If we take:

             7 x 7

we can say 7 x 7 is 49 in normal arithmetic. 49 divided by 8 is 6 remainder 1 and it is the remainder that tells us the answer.  So we can say:

             7 x 7 = 1

This is in modulo 8, remember. If we were using a different modulo, the answer would be different too.

So now you can complete the times table for modulo 8 and, again, I have put on the ones we have already done:

What patterns can you see?

Just to help you, the answer is at the end of this page, but don’t look yet!

Now you are away. You can make up times tables for any module you like.

We could do a Modulo 9 times table, for instance. This would have the numbers 0 to 8.

Can you work out by counting round the circle or by using the remainder idea what

          7 x 4

comes to in Modulo 9?

The answer is 1. Did you get it?

The other thing you may have realised by now is why modulo arithmetic is sometimes called ‘Remainder Arithmetic’.

Have some fun making up your own Modulo times tables and tell your friends and teacher. It’s great fun.

Surprisingly, Modulo arithmetic does have a lot of uses in computers and other places. We use it all the time with our clocks, of course (except we put the number 12 where the number 0 should go, but the idea is the same). Have fun!

Answers

Here are the answers to the Modulo 6 and Modulo 8 multiplication squares, just to check you have the right idea, but don’t look until you have tried them yourself.

Can you see all the patterns that appear?