Lesson video

Hiya.

My name is Ms. Lambel.

Really pleased that you've decided to pop along and do some maths with me today.

Welcome to today's lesson.

This lesson is in the unit of property of numbers, and the title of today's lesson is "Expressing an integer as a product of its prime factors." So by the end of this lesson, you'll be able to write any positive integer uniquely as a product of its prime factors.

Now remember, learning sometimes can be challenging, but don't worry, I'll be here with you every step of the way to guide you and help you through.

So here are some key words that we will be using throughout today's lesson, so let's just have a look at those.

Composite number.

A composite number is an integer with more than two factors.

All integers greater than 1 are either composite or prime.

Prime factors.

Now, these are the factors of a number that are themselves prime.

And the other one is unique.

Each composite number has a unique product of its prime factors; i.

e.

, there is only one unique way it can be written.

Some other words that we will be using throughout today's lesson which you should be familiar with are prime number, so remember that is a number that is greater than 1 with exactly two factors, and product is the result of two or more numbers X together.

So today's lesson we're gonna split into three learning cycles.

So the first one, Composite and prime numbers, second one, Product of prime factors, and then finally, we'll finish up with Product of prime factors using index notation.

Okay, so let's get going firstly with Composite and prime numbers.

So Alex is using multiplication and the first three prime numbers to create the numbers from 1 to 20.

So he's using the first three prime numbers, which are 2 3 and 5.

Here are the numbers 1 to 20.

So what we'll do is we're gonna go through each of these and we're going to decide if we can use multiplication and just those three numbers to create the numbers 1 to 20.

So number 1, no, we can't, okay? The only way we can make 1 as a product is 1 X 1.

2.

Okay, we have got a 2 there, but we can't do a multiplication with 2 as the only thing we could do would be 1 X 2 and 1 is not one of the numbers that we are using.

Okay, same with 3.

So 3 is there, but we haven't got the 1 to prepare with it to make that product.

4.

So 4 can be written as 2 X 2.

5 exactly the same as 2 and 3.

6, so 6 we could write as 2 X 3.

7 is the same as 2 3 and 5.

8.

Remember here, we're only using numbers 2 3 and 5, but we can make 8 by doing 2 X 2 X 2.

9 we can create from 3 X 3.

10, 2 X 5.

11.

No, we can't make that one.

12 would be 2 X 2 X 3.

13.

Again, we can't make that one.

14.

No, because I would need another number there, so not that one.

15 we can make from 3 X 5.

16, 2 X 2 X 2 X 2.

17 we can't do.

18 would be 2 X 3 X 3.

19 we can't do, and 20 we can do.

It would be 2 X 2 X 5.

So Alex has been through each of these and decided if he can use multiplication and the numbers 2 3 and 5 to create them.

These are the ones he can do.

So here's a list of the numbers that he couldn't create: 1, 2 3 5 7, 11, 13 14, 17, and 19.

So Alex has spotted something.

He spotted that all of those numbers are prime apart from 1 and 14.

Sophia says, "I think you could make more numbers if you use the first four prime numbers." So Sophia's adding in that actually maybe here we could, if we used another prime number, be able to make more of them.

So is Sophia right? Okay, she is because we can do 2 multiply by 7 to make 14, and remember 7 would be the next prime number.

So numbers greater than 1 are either composite or prime.

So notice here greater than 1.

That's really important.

Composite numbers have more than two factors.

Prime numbers have exactly two factors.

So that's how we distinguish between the two.

Composite numbers have more than two factors, and prime numbers have exactly two factors.

So check for understanding.

True or false.

So 1 is a composite number.

So I'd like you please to pause the video, decide whether you think the answer is true or false, and then give your justification.

Super.

Let's see how you got on with that question.

So 1 is a composite number, true or false? Well, that's false.

So remember, a composite number has to be greater than 1, okay? And also, it has to have more than two factors, so it was b, okay? It's not a prime number, so it must be composite.

Remember, that's not true.

Composite numbers have to be greater than 1 and they must have more than two factors.

Here's our first task.

So what I'd like you to do, you should be familiar with what an array is.

Remember, that is a sequence of dots that we use to create these numbers.

So I would like you to draw arrays to prove that the following are composite numbers.

So you're gonna pause the video and then you're gonna come back when you've got your answers drawn out for those.

Brilliant, well done.

Let's check those answers then.

So for the first one, okay, we can prove that 18 is a composite number by showing this array, so a 3 by 6 array.

Remember, this is just an example.

You may have drawn a different array, sorry.

Okay, 32 I've done a 8 by 4, 25 5 by 5, and 51, I've done a 3 by 17.

So we're gonna move on now to our next learning cycle: Product of prime factors.

So we know what a product is, we know what a factor is, and we know what a prime is, so we should be able to whizz through this no problems. So earlier, we tried to make composite numbers up to 20 using prime numbers.

What would happen if we were to look at larger numbers? So here's a reminder of the first eight prime numbers: we've got 2 3 5 7, 11, 13 17, and 19.

We're going to start with 21, so can we use two of those numbers or more than two of those numbers to create the product 21? Yes, we can.

That would be 3 X 7.

What about 22? Have a think about that one.

Hopefully, you spotted that could be 2 X 11.

And now, 23 have a think about that one.

And you should have come up with the fact that actually we can't do that one.

Why can't we do 23? Have a think about why we can't do 23.

Can't do 23 because 23 actually is a prime number itself, isn't it, if we remember back to when we looked at prime numbers.

So here's Alex and Sophia again.

So Alex says, "Does that mean that all composite numbers can be written as prime factors?" Okay, I want you to think about that question first.

Sophia says, "It does.

"All composite numbers can be written as a unique product of prime factors." So unique, remember that? It can only be written one way.

So let's take a look.

Alex and Sophia decide they're going to test this out, so they're both looking at the number 30.

Let's look at what Alex did.

So Alex has said, "I know a factor pair of 30 are 10 and 3 "and I know that 10 is 2 X 5, and then I just multiply that by 3." Sophia decides to start with a different factor pair of 30, so let's see what happens when she does that.

So she's chosen 6 and 5 so 6 X 5.

But she knows that 6 is 3 X 2 X 5.

So notice that even though they both started with a different factor pair, they ended up with the same prime factors because it's unique, it can only be written in one way.

Let's have a go at this one together, so we are moving on now to a much bigger number.

But don't worry, it's actually pretty straightforward because I know you're really, really good at factor pairs.

So we've got 1,260.

I know that that is 126 X 10.

Remember, you can start with any factor pair you like.

So I'm now looking to break those two numbers down.

So 126 I've decided to write as 2 X 63.

So I spotted it was even, thought, "Okay, let's go with the fact that we know that then 2 is a factor," and then I halved it.

And then 10, okay, is 2 X 5 right? So we need to have a look and see whether we can break any of these down further, and we can: 63.

So two numbers that multiply together to make 63, 7 and 9 I've gone for.

Now, we need to repeat that.

Can we break any of those down anymore? Yes, we can.

We can break the 9 down.

So the 9 is actually 3 X 3.

Can we break that down any further? No.

So all I've done because we like to be organised as mathematicians is I've just then rewritten those in numerical order: so 2 X 2 X 3 X 3, X 5 X 7.

Okay, let's have a go at another one of those together and then I'm gonna give you an opportunity to try it for yourself.

So 210.

So factor pair, here I've gone for 21 and 10, so I'm breaking down now the 21 and the 10.

So 21 I know is 3 X 7 and 10 is 2 X 5.

Can any of those be broken down anymore? No.

And again, I've just tidied it up and I've written those in numerical order: 2 X 3 X 5 X 7.

So what I'd like you to do now is to have a go at writing 315 as a product of its prime factors.

So pause the video and come back and check in with me when you're done, and we'll see how you got on.

Brilliant.

Super well done.

So let's see how you got on with that.

Remember, you might not have started with the same factor pair as I have.

Okay, that's absolutely fine.

As long as you've made no errors, we'll end up with exactly the same answer.

So you should have got, I did 315 is 5 X 63, then I know that 63 is 7 X 9, and then 9 is 3 X 3.

Moving on now then to Task B.

So Task B, you're going to do exactly what you've just done there.

You're gonna write these four numbers as a product of their prime factors.

So just doing exactly the same thing as we just did on the previous slide, and when you are done, you can come back and check your answers.

So pause the video now and then have a go at these and come back when you're ready.

Superb work.

Well done.

Let's check your answers.

So, a, 78, remember you might've started with a different factor pair; that's absolutely fine, but your answer is going to be the same as mine.

So 78 is 2 X 3 X 13.

b, 165 is 5 X 3 X 11.

525 is 3 X 5 squared X 7.

Don't worry if you've not written your answer like that; I've just slipped that in.

We're gonna come back onto that in a moment in the next learning cycle.

So you should probably have written 3 X 5 X 5 X 7.

And then, d, 840 is 2 X 5 X 2 X 2 X 3 X 7.

But I'm sure you've probably written that as 2 X 2 X 2 X 3 X 5 X 7 because remember I said we like to be organised as mathematicians and write them in the correct order.

I've not done that here, so it doesn't matter.

Just remember that the working out steps are only examples.

The final line should be the same.

The only thing that we can do is change the order of those prime factors.

So we're now gonna move on to our final learning cycle.

Our final learning cycle, and I've just sort of touched on it a little bit in one of those answers, is writing our answers using index notation.

So let's move on and have a look at what that's going to look like.

So here they are again, Sophia and Alex, and they've been practising writing numbers as a unique product of their prime factors.

Sophia says, "300 written as a product of its prime factors is 2 X 2 X 3 X 5 X 5," and Alex says it's 2 squared X 3 X 5 squared.

So who is right? They're actually both right.

So Alex has just written his answer using index notation.

So he's used index notation, so he's spotted that 2 X 2 is actually 2 squared and 5 X 5 is actually 5 squared.

So he's just written exactly the same thing but in a slightly different way, so using that index notation.

So just to recap something that we've come across before, index notation is when we write a repeated multiplication using a base and an exponent.

So here, the 7 is the base and the 4 is the exponent, so remember that means a repeated multiplication of 7 four times.

Now, let's have a check and see how you've got on with your understanding of that.

So which of the following shows 2 X 2 X 3 X 3 X 5 X 7 written as a product of their prime factors using index notation? So pause the video, have a think, and then come back and see how you got on.

Super well done.

Let's have a look and see how you got on.

I'm sure you got it right.

So it was actually the final one.

So if we looked, we had 2 X 2 so that's 2 squared, so we couldn't rule a out at that point.

Then, we had 3 X 3 which was 3 squared, so we then knew that it couldn't be b.

And then we had 5 X 7, and on the top one they've just actually written that as its product of 35.

So it was c.

Well done if you got that right.

Don't worry if you didn't.

Just make sure you understood all of those reasons I've just given.

So 360 is 2 cubed X 3 squared X 5.

We can use this to write other numbers as a product of prime factors, but we don't need to repeat the process all the way from the beginning.

So let me show you what I mean by that.

720.

Well, we know that that is actually 360 X 2, but we know that 360 as a product of its prime factors is 2 cubed X 3 squared X 5.

We already knew that, so therefore we can replace the 360 with a product of its prime factors and then we multiply that by 2.

I've now got an answer of 2 to the power of 4 X 3 squared X 5.

Why has the power of 2 changed? Well, previously I had 2 cubed, but I've now got another 2 so I've done another repeated multiplication of 2, so it's 2 to the power of 4.

Let's look at another one.

So 3,600.

Well, remember we're using 360, so 360 X 10 gives us 3,600.

But we know that 360 is that as a product of its prime factors, and then we're gonna multiply that by 10, but we know that 10 as a product of its prime factors is 2 X 5.

Again here, we're just gonna tidy that up.

So I've got a 2 cubed and I've got a 2, so it's gonna give me 2 to power 4, I've got a 3 squared, and now I've also got a repeated multiplication of 5 twice, so that's 5 squared.

So once we've tidied that up, it's 2 to the power 4 X 3 squared X 5 squared.

Let's check our understanding of what we've just looked at.

Which three numbers can be found using 12 written as a product of its prime factors? So one of them can.

Pause the video and have a think about it.

Great work.

So let's have a look.

Can 2,400 be found? Yes, it can.

12 is a factor of 2,400.

Can 48 be found? Yes.

Again, 12 is a factor of 48.

Can 90 be found? No, 12 is not a factor of it.

But 360 could be done because 12 is a factor of 360.

Well done if you got all of those right.

Super well done if you've also started to think about the product that you would use.

Now, we can actually check our answers using our calculators.

So here is a Casio fx-991EX.

So I'm gonna show you how to do it on this calculator, and your calculator might not be the same, but there will be a function on it that allows you to check your answers using the Prime Factor button.

So firstly, we need to ensure that "Calculate" is highlighted; so we can see here that we've got "Calculate" highlighted.

We then need to press the EXE button, so that's in the bottom right-hand corner.

So we're going to check, so 360 we've used a few times, so we're just going to check that we've been using the right product to prime factors.

So we are going to type in 360 and we're gonna press the EXE button.

We're then going to click the button that says FORMAT.

So it's next to the EXE button.

If you've got the same calculator as me, it's there highlighted at the bottom for you.

We then need to go down.

We are doing prime factors, so pretty obvious here, we're gonna go scroll down and get to the Prime Factor.

And then, when we've done that, we're gonna press that EXE button again.

And we can see here that we've been using the right product to prime factors all along, so 2 cubed X 3 squared X 5.

So it's worth remembering so that you can check your answers this way.

Also, if you've got a different calculator, you'll be able to look up how to do it on your calculator.

So like I said, we've just checked that those things are the same.

Now, on to our final task of today's lesson, so well done.

You've done so well so far.

Let's keep it going just for the last little bit.

So Task C, we are going to fill in the missing digits.

So in the blue boxes, I'd like you to decide what digit needs to go in there.

So you're gonna pause the video now and when you're done, you're gonna come back and you're gonna check your answers.

I'm pretty certain you're gonna have got them all right, but do come back and make sure that you've got all of the correct answers.

Right, in this task, there is a second question.

Now, notice I've put in blue there "without using a calculator." I know I've just shown you how you can use your calculator to find prime factors, but remember that was just to check your answers.

So here we want to use the fact that 60 equals 2 squared X 3 X 5, and we're going to use that technique I've just shown you to work out what these numbers are using that fact.

So you're gonna pause the video and when you're ready, come back and we'll check in with those answers.

Wow, that was quick.

Let's check our answers then.

So we should have for question number 1, your missing digits were 3, 2, and 2.

Sorry, 2 cubed X 3 squared X 7 squared.

b, the missing digits were 2, 3, 2, so 2 squared X 5 cubed X 11 squared.

So remember, we are just looking at how many times we've repeated the multiplication.

c is 3 squared X 5 to the power of 4 X 13 squared.

d, 2 squared X 7 cubed X 17 to the power of 4.

And then, e, 2 X 5 cubed X 7 cubed X 11 squared.

And then, f, why was there no box for the 2 in e? So the index would be 1, and we don't write an index of 1.

Remember that.

Now, we're moving on to question 2.

So, a, the final answer you should have is 2 cubed X 3 X 5, b, 2 squared X 3 squared X 5, c, 2 squared X 3 X 5 squared d, 2 cubed X 3 X 5 squared e, 2 to the power of 4 X 3 X 5 cubed, f, 2 cubed X 3 X 5 squared X 7.

And well done if you got the answer to g right.

So I really challenged you there; I put one in where actually you were doing a division.

So 60 divided by 2 gives us the 30.

We know that was 60 was 2 squared X 3 X 5, but we needed to divide that by 2, and 2 square divided by 2 just leaves us with 2.

So the answer was 2 X 3 X 5.

Absolutely superb, well done if you got that one right.

I just thought you'd be up for a challenge right about now.

Now, we'll summarise our learning.

Numbers greater than 1 are either composite or prime.

Composite numbers can be written as a unique product of prime factors.

So for example, 52 is 2 X 26, but 26 is 2 X 13.

So we could write it as 2 X 2 X 13.

And products of prime factors can be written more efficiently using index notation.

So instead of writing 2 X 2, we could write 2 squared.

So example, 52 equals 2 squared X 13.

Well done.

You did so well today.

Really impressed with what we've managed to achieve.

Well done.