Lesson video

Hi, my name's Mr. Peters.

Thanks for joining me today.

This is the beginning of a brand new unit and in this unit we're gonna be thinking lots about common factors and multiples and how we can move towards using these to help us calculate a lot more efficiently.

In this lesson today, we're gonna be thinking about finding common factors of two or more numbers.

If you're ready to get started, let's get going.

So, by the end of this lesson today, you should be able to say that, "I can explain how to identify common factors of two or more numbers." Throughout this lesson today, we've got a number of key words that we're gonna be referring to.

I'll have a go at saying them first and then you can repeat them after me.

The first one is factor.

Your turn.

The second one is common factor.

Your turn.

And the last one is prime factor.

Your turn.

Let's think a little bit more about what these mean.

So, a factor is a whole number which exactly divides another whole number.

When we compare lists of factors for any two numbers, any factors that are the same are said to be common factors.

And finally, a prime factor is a factor of a number that is a prime number.

This lesson today is broken down into two cycles.

The first cycle, we'll be thinking about finding common factors, and the second cycle, we'll be thinking about using prime factors to find common factors as well.

Let's get going.

Throughout this lesson today, we're also gonna be meeting Jun and Alex who'll be sharing their thinking as we go throughout the lesson.

Okay, so hopefully you can see here we've got two factor bugs and these factor bugs are representing the factors of numbers 18 and 30.

Jun says that it looks like some of the factors on both 18 and 13 are the same.

Let's have a look.

Can you take a moment and have a look for yourself? Can you see any factors that are the same in both numbers? Jun says that he knows that both numbers have number 1 as a factor.

Both numbers have number 2 as a factor.

They also have number 3 as a factor, and they have number 6 as a factor.

Alex says, we call these common factors.

We can say that the common factors of 18 and 30 are 1, 2, 3, and 6.

Take a moment to have a think.

Which one of those factors could we call the highest common factor? That's right, we could say that 6 is the highest common factor.

It's the largest factor that is common between both numbers.

And, again, have a quick think.

Which number would we say is the lowest common factor this time? That's right.

1 is the lowest common factor, isn't it? It's the smallest factor a number can possibly have, and that number 1 is shared between both 18 and 30.

We can use Venn diagrams to help us sort some of these factors out.

Have a look at the diagram in the middle.

The circle on the left represents factors of 18, and the circle on the right would represent factors of 30.

Where these two circles overlap in the middle is a position where we can place factors that are common for both numbers.

So, let's go through these then, shall we? We can say that 18 and 9 are two factors of 18.

These are factors which aren't factors of 30.

So these would go in the part of the circle on the left-hand side.

We can say that 30, 15, 10, and 5 are factors of 30, which aren't factors of 18.

And, again, these would then therefore go on the part of the circle on the right-hand side.

So, which ones were our common factors? That's right.

1, 2, 3, and 6 were our common factors, weren't they? And they go in the middle between both of the circles where both of the circles overlap.

Again, we can therefore say that the common factors of 18 and 30 are 1, 2, 3, and 6.

Could you have a go at saying that? Well done.

Let's have a look at another example now.

This time we've got numbers 16 and 32.

Again, take a moment to have a think for yourself.

Can you identify any common factors between these two numbers? That's right.

Number 1 is a common factor.

Number 2, again, is a common factor.

Number 4, this time, is a common factor.

Are there any others? Yep, 8 is a common factor and 16 is also a common factor.

We can say, therefore, that the common factors of 16 and 32 are 1, 2, 4, 8, and 16.

Again, could you say that with me? The common factors of 16 and 32 are 1, 2, 4, 8, and 16.

Take a moment to have a think again then.

This time, what's the highest common factor? That's right.

16 is the highest common factor.

And what would the lowest common factor be? That's right.

1 is the lowest common factor again, isn't it? Let's place these into our Venn diagram now then.

We're gonna place 32 in the right-hand side 'cause we know 32 is not a factor of 16.

And then, all of the other factors, we're gonna place in the middle here.

What do you notice? That's right.

The common factors of 16 and 32, as we know, were 1, 2, 4, 8, and 16.

And actually, there are no factors that belong only to 16 and not to 32.

Therefore, every factor of 16 is also a factor of 32.

I wonder why that is.

Hmm, I wonder if it's anything to do with the fact that 32 is double 16.

Therefore, every factor of 16 can also be found to be a factor of 32.

I wonder if that works for other numbers that are doubled.

Maybe that's an investigation you could look into for yourself.

Here's one more example for us to look at.

This time, we've got number 7 and 11.

Have a look at the factors.

What do you notice this time? There's only one common factor, isn't there? And that common factor is number 1.

Let's say this together.

The common factor of 7 and 11 is 1.

Alex has pointed out something really interesting here.

He's noticed that 1 was a common factor for all of the examples we've looked at so far.

So we can say that 1 is a common factor of every number.

Let's have a look at that in our Venn diagram again.

On the left-hand side in the circle, we've placed a 7 'cause 7 is not a factor of 11, is it? We also know that 11 is not a factor of 7, so we'd place that on the right-hand side.

And then in the middle, we'd place 1 'cause we know 1 is a common factor of both 7 and 11.

Okay, time for you to check your understanding now.

Can you find the common factors of 8 and 14? You might like to draw them as a factor bug on a piece of paper if that helps you.

Take a moment to have a think.

Okay, so, the factors of 8 were 1, 2, 4, and 8.

And the factors of 14 were 1, 2, 7 and 14.

And we can see that the common factors were both 1 and 2.

And we can place the factors into our Venn diagram just like this.

Hopefully, you were able to do that too.

We can say, therefore, that the common factors of 8 and 14 are 1 and 2.

Another quick check for understanding then.

True or false? Two numbers can have exactly the same factors.

Take a moment to have a think.

That's right.

It's false, isn't it? And have a look at these justifications here.

Which one of these help you to reason your thinking? That's right.

It's B, isn't it? B says that each number has itself as a factor, therefore, where if you think back to 16 and 32 from earlier on, 32 was a factor of 32, but it wasn't a factor of 16 because 32 is a much larger number than 16, and, as we know, every number has itself as a factor, therefore, we can never actually have two numbers that have exactly the same factors.

Okay, time for you to have a go at practising now.

For Task Number 1, what I'd like you to do is find the common factors of the following numbers.

For Task 2, I'd like to think about what the highest common factor is for these two numbers.

And for Task 3, I'd like to think about what the lowest common factor is for these two numbers.

And then, for Task 4, can you find two pairs of numbers where the highest common factor of the first pair is the second highest common factor of the second pair? How many different solutions are you able to find? And an extra task for Task 5 here, what I'd like to do is use the digits 0 to 9 only once.

And here, you need to fill in a number for each of the boxes where you're gonna create two numbers and then underneath that, you're gonna find a number using those digits that's a common factor of both of the numbers above it.

Good luck with that and I'll see you again shortly.

Okay, welcome back.

Let's run through these answers then.

So, the common factors of 15 and 30 were 1, 3, 5, and 15.

The common factors of 12 and 36 were 1, 2, 3, 4, 6, and 12.

And the common factors for 25 and 100 were 1, 5, and 25.

For Question 2, the highest common factor of 19 and 38.

Well, the only common factors of 19 and 38 are 1 and 19, therefore, the highest common factor is 19.

And now we're looking for the lowest common factor of 14 and 42.

Well, the common factors of 14 and 42 were 1, 2, 7 and 14, and yet, therefore, the lowest common factor would be 1.

In fairness, we didn't even need to work out the other common factors, did we? We know that 1 is always the lowest common factor of any two numbers.

Okay, for Question 4 then, let's have a look at an example of this.

We were looking for two pairs of numbers, weren't we? The highest common factor of the first pair would be the second highest common factor of the second pair.

Let's have a look at an example we come up with.

So, for 8 and 12, we could say that the common factors are 1, 2, and 4.

And 4 would be the highest common factor.

So now we're looking for two numbers where 4 would be the second highest common factor.

Well, here's an example, 16 and 24.

The common factors of 16 and 24 were 1, 2, 4, and 8.

And therefore, we can see that 4 is the second highest common factor in this example.

Well done if you managed to find an example of your own as well.

And then, for the last task then.

Using the digits 0 to 9 only once, were you able to find two numbers and a common factor of both of those numbers? Here's an example that I've come up with.

16 and 32 were the two numbers that we had and the common factor of that was 8.

I was a bit cheeky there and I placed a 0 in the tens column, didn't I? For my common factor.

I wonder if you managed to find a different solution.

Some of your friends might have as well.

Share your different solutions.

Were there any strategies that you noticed as you went along? Well done for completing all of that.

Okay, moving on to Cycle 2 now then, we're gonna be thinking about using prime factors to help us find common factors.

Let's take a moment to remind ourselves about prime factors then.

We know that every number can be composed multiplicatively of prime factors.

Here's an example, we've got 18 here, and we can see that 18 has factors of 2 and 9.

2 is a prime factor, isn't it? And that's represented on our circle here by the yellow segment.

We know that 9, the other factor, is not a prime number, therefore, we need to break 9 down into its prime factors.

We know that 9 is composed of 3 multiplied by 3.

And 3 is a prime number.

So, therefore, we can say that 3 and 3 would also be prime factors of 18.

And they are represented on our circle by both of the green segments.

So, we could say that 18 is equal to 2 multiplied by 3, multiplied by 3.

We can now use prime factors to find the factors of a number.

So, whilst we know that the prime factors of 18 are 2, 3 and 3, we could multiply the 2 and the 3 together to find a factor that would, therefore, be 6.

We could multiply the 3 and the 3 together to find a factor and that, again, would be 9, wouldn't it? So, we've now found all of the factors of 18.

We know that the prime factors of 18 were 2 and 3 and we've also found two other factors of 18, 6 and 9, haven't we? We also know that every number has 1 and itself as a factor, therefore, 1 and 18 would also be factors of 18.

I wonder how we could apply this now to thinking about finding the common factors of 2 numbers.

Well, let's have a look.

We've got 18 and 30 here and we've broken 18 and 30 down into its prime factors.

The prime factor composition of 18 is 2 multiplied by 3 multiplied by 3.

And the prime factor composition of 30 is 2 multiplied by 3, multiplied by 5.

We know that because the yellow segment represents 2, the green segment represents 3.

If we multiply those two together, that would give us 6, and then we multiply that by the blue segment, which would be 5.

Therefore, altogether 6 times 5 would be equal to 30.

So, how could we represent this in our Venn diagram then, thinking about finding the factors of all of these numbers? Well, we can represent the common prime factors, can't we, between these two numbers? Both of these numbers have 2 and 3 as a prime factor.

So we can place these into the middle of our Venn diagram.

That would leave us with a 3 on the left-hand side, wouldn't it? And that would leave us with a 5 on the right-hand side.

So, so far we can say that 2 and 3 are common factors of 18 and 30.

We can then think back to what we just did previously with our prime factors.

We can now multiply these two common prime factors together to find other common factors.

So we know that 2 multiplied by 3 is equal to 6, so therefore, 6 is also a common factor of 18 and 30.

We can say that the common factors of 18 and 30 are 2, 3, and 6.

Let's have a look at another example, shall we? Here are the numbers 12 and 48, and they've been broken down into their prime factor compositions.

The prime factor composition of 12 is 2 multiplied by 2, multiplied by 3.

And the prime factor composition of 48 is 2 multiplied by 2, multiplied by 2, multiplied by 2, multiplied by 3.

So, take a moment to have a think.

Can you see any common prime factors here? That's right.

We can see that there are two sets of 2 and one set of 3 that are common in the prime factor compositions of both 12 and 48.

We'll place these in the centre of our Venn diagram.

Well, there are no more prime factors left in our composition of 12, however, there are two sets of 2 as a prime factor left from the composition of 48.

So, we can say that 2, 2, and 3 are common prime factors of 12 and 48.

And, as we know, if we start multiplying these common prime factors together, we can find other common factors.

So we could do 2 multiplied by 2, that would be equal to 4.

And we could do 2 multiplied by 3, which would be equal to 6.

We could also do 2 multiplied by 2 multiplied by 3, which would be equal to 12, as we know from the first one.

So 12 would also be a common factor of both 12 and 48.

We can say in total that the common factors of 12 and 48 are 2, 3, 4, 6 and 12.

And, of course, we know that 1 is a common factor of every number, so we could also include 1 in that list as well.

Okay, time for you to check your understanding now.

Can you tick the prime factors of 18? Take a moment to have a think.

The prime factor composition of 18 would be 2, multiplied by 3 multiplied by 3.

Therefore, 2 and 3 are prime factors of 18.

And another quick check then.

24 can be expressed as prime factors as 2 multiplied by 2, multiplied by 2, multiplied by 3.

And we can see that here below.

Can you use these prime factors to find the factors of 24? Take a moment to have a think.

Well, as Alex has pointed out, we know that 1 and 24, of course, will be factors of 24.

And we could then start multiplying these prime factors together.

2 multiplied by 2 is equal to 4.

2 multiplied by 3 is equal to 6.

2 multiplied by 2, multiplied by 2 would be equal to 8.

And 2 multiplied by 2, multiplied by 3 would be equal to 12.

So, the factors of 24 would be 1, 2, 3, 4, 6, 8, 12, and, of course, 24.

Well done if you managed to get that.

And time for you to practise our second task for today then.

What I'd like to do is use common prime factors to find the common factors of the following pairs.

Once you've done that, we're going to look to find a number.

We've got the information that the prime factors of the number are 2, 3, and 7.

Once you've found the number, can you identify the other factors for that number as well? Good luck with those two tasks.

When you're finished, come back and we'll go through them together.

See you shortly.

Okay, welcome back.

Let's run through these together then.

So, the common prime factors of 27 and 36.

Well, throughout those compositions, both of those numbers have 2 sets of 3.

So to find the common factors of 27 and 36, we can say that 3 is a common factor, can't we? 'cause that's a common prime factor.

And then we can multiply these two together and that would give us 9.

So 3 and 9 are the common factors of both 27 and 36.

What about 15 and 45 then? So we can see that both those numbers have 3 and 5 as common prime factors in their composition.

So we can say that 3 and 5 are common factors and then we can multiply these two together again, which would be 15.

So, 3, 5, and 15 are the common factors of 15 and 45.

And then finally, 8 and 48.

Have a look carefully.

There are three lots of 2 in their composition of prime factors.

So we can say that 2 is a common factor of 8 and 48, and then we could start multiplying these together as well.

So 2 multiplied by 2 would be 4, and then 2 multiplied by 2, multiplied by 2 would be 8.

So we can say that 2, 4, and 8 are also common factors of 8 and 48.

And, of course, we know that 1 is a common factor of all of these as well.

Well done if you managed to get all of those.

And for Task 2, how did you work out the number? Well, we know that we could multiply each of these prime factors together to find the number, couldn't we? So 2 multiplied by 3 is equal to 6, and then if you multiply that by 7, that would give us 42.

So the number is actually 42.

And then, starting to think about the other factors of 42, well, we know that 1 and itself, of course, are always factors.

So 1 and 42 would be factors.

And then we can start multiplying these prime factors together.

2 multiplied by 3 is equal to 6.

2 multiplied by 7 is equal to 14.

And 3 multiple by 7 is equal to 21.

So therefore, 6, 14 and 21 would also be factors of 42.

Well done if you've got that as well.

Okay, that's the end of our learning for today.

Hopefully you've enjoyed yourself and can think more flexibly now about how to find factors and common factors of numbers.

We can summarise today's learning by saying that where two numbers have the same factors, these are common factors.

We can represent common and uncommon factors using a Venn diagram.

And you can find the factors of a number by multiplying some of its prime factors together as well.

Thanks for joining me today.

I really enjoyed that lesson.

And hopefully you did too.

Take care, and I'll see you again soon.