Lesson video

Hiya, my name is Ms. Lambell.

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 properties of number and the title of today's lesson is Calculating integers from their prime factor expressions.

By the end of this lesson, you'll be able to evaluate a number written as a product of its prime factors.

There are some key words that we will be using throughout this lesson.

A composite number is an integer with more than 2 factors.

All integers greater than one are either composite or prime, product is a result of 2 or more numbers being multiplied together.

And prime factors are factors of a number that are themselves prime.

So you should be familiar with all of those words.

We're also going to be looking at numbers written in their exponent forms, so remember we've got the base.

So in this example here, the base is 2 and the exponent is 3.

And the exponent represents a repeated multiplication and how many times we repeat that multiplication.

Today's lesson, we are going to be spitting into 2 learning cycles, the first of which is going to be evaluating product of prime factors, and then we'll move on to using product of prime factor form.

So let's start with the first one.

Evaluating products of prime factors.

Right, let's get going.

All composite numbers can be written as the product of a unique combination of prime numbers.

You should be aware of that already.

So Lucas and Aisha have worked out which number is represented by 2 to the power of 5, multiplied by 3, multiplied by 5 cubed.

So Lucas says it's 60,000 and Aisha says it's 12,000.

Can they both be right? No, they can't both be right, because remember, each composite number has a unique product of prime factors.

It can only be one of those.

Let's move on to this check for understanding.

So it's a true or false, 2 cubed multiplied by 3, multiplied by 5 squared equals 600.

So do you think that is true or false? And then also pick your justification.

So because 8 multiplied by 3 multiplied by 25 is 600 or because 6 multiplied by 3 multiplied by 10 equals 600.

So pause the video, decide whether it's true or false, and then choose the correct justification.

Let's see how you got on with that.

So it was true, so yes it was true.

And the correct answer was A, the justification because 8 multiplied by 3, multiplied by 25 is 600, or 2 cubed is 8, 5 squared is 25.

We've multiply those together, we get 600.

Okay, then guys, over to you now.

So this is our first task within this lesson and what I'd like you to do is to write the following as integers.

So you can use your calculator.

Make sure to try and use the exponent button rather than the just typing in repeated multiplication 'cause remember, mathematicians are all about being efficient.

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

Super, well done.

Let's have a good look and we'll check those answers then.

So here are our answers.

Remember it's a unique product of prime factors.

So this is the only answer it could be.

So one was 1,800, 2 was 875, 3, 1,452, 4, 31,941 and 5 was 162,925.

So give those a check.

Make sure you've got all of those right.

If you didn't, it was probably just a slip on your calculator.

So it's always worth check it doublechecking that what you think you've typed into the calculator you have.

Let's move on now to our next learning cycle.

So we're going to be looking at using product of prime factor forms. We're actually going to be using it.

So using numbers written as a product of their prime factors.

We can determine other facts without evaluating the number.

2, multiplied by 2, multiplied by 3, multiplied by 3.

Is this a square number? So remember a square number is the product of 2 identical integers.

So we could rewrite this as 2 multiplied by 3, multiplied by 2, multiplied by 3.

We've then got 2 identical integers.

So yes, we could write that as a repeated multiplication of 2 multiplied by 3 to the power of 2 or squared I should say.

So yes, that is a square number.

Let's have a look at this number now.

So we've got 2 squared multiplied by 3 multiplied by 11 squared.

Is this an even number? What do you think? How do we know if that's gonna be an even number? Remember we're not typing this into our calculator and writing it as an integer.

We're using that prime factor form.

So all even numbers remember are multiples of 2.

And since 2 is one of the prime factors, it must be a multiple of 2 and therefore it must be even, maybe you spotted that for yourself.

Well done if you did.

So we're sticking with the same product primes.

Now we need to think about is it a multiple of 4? So is that a multiple of 4? A multiple of 4 must have a factor of 4.

So does that product of primes have a factor of 4? Well, yes it does because 2 squared is 4.

So therefore this must be a multiple of 4.

So we're still sticking with that same prime factor form.

Now want you to have a think about is this a multiple of 10? Is it a multiple of 10? Well, what do we know? Remember again, we are not putting this into the calculator and working out what it is as an integer.

Okay, we are just saying can we work it out just from what we've got on in its current form.

So 10 is 2 multiplied by 5 as a product of its prime factors.

Does our product of prime factors have 2 multiplied by 5 in it? No, it has a 2 but it doesn't have a factor of 5.

So therefore it can't possibly be a multiple of 10.

What I'd like you to do now is to think about could we make it a multiple of 10? Well, I'll answer that one for you, yes we can.

So I'd like you just to think about how could we make that a multiple of 10.

We need to introduce a factor of 5 because we've said that 10 as a product of its prime factors is 2 multiplied by 5.

Our product of prime factors that we are working with already has a factor of 2.

So we can introduce a factor of 5 and that will make it a multiple of 10.

So here I've got 2 squared multiplied by 3, multiplied by 5, multiplied by 11 squared.

That would be a multiple of 10.

That's not our only answer 'cause remember we can find other multiples.

We're now ready for a check for understanding.

Which two of the following are multiples of 10? So I've given you 4 product of primes and 2 of those are multiples of 10.

Now remember we're not going to get our calculators out, we're not gonna type them in and see if there's a zero in the ones column.

We are using the knowledge that we've got about multiples of 10 and prime factor form.

Pause the video and come back when you are ready.

Well done, I hope you've managed to identify which 2 are the correct multiples of 10.

So let's go through them in turn.

So the first one, remember we know that multiples of 10 must have a factor of 2 and 5.

So the first one cannot be a multiple of 10 because it has a factor of 2, but not 5.

Let's look at b.

This one, it has a factor of 2 squared, which means it must have a factor of 2 and it has a factor of 5 squared, which means it must have a factor of 5.

So yes, it's a multiple of 10.

Now let's look at c, so we've got 2 squared.

So if 2 squared is a factor, then 2 must be a factor.

We've got 5 cubed.

Well, if 5 cubed is a factor, then 5 must be a factor.

So yes, that is a multiple of 10.

And the final one, we have not got a factor of 2, so therefore it's not a multiple of 10, of course, remember here if we wanted to make it a multiple of 10, we could just introduce that factor of 2.

Earlier we looked at whether something was a square number.

Remember a square number is the product of 2 repeated integers.

I think the example we had was 2 multiplied by 2, multiplied by 3 multiplied by 3.

And we rearranged that to give us 2, multiplied by 3, multiplied by 2, multiplied by 3.

So yes, it was a square number.

So this one's a little bit more complicated, this one because we've got some exponents that we need to be considering.

So I could rewrite this as 2 multiplied by 3, multiplied by 11 squared, multiplied by 2, multiplied by 3, multiplied by 11 squared.

So have I actually found the product of 2 repeated integers? Yes, this is a square number.

Remember a square number is a product of 2 repeated integers and we have here 2 repeated integers.

So it is a square number.

Right, so I'm not gonna ask you whether this is a square number because you'll have noticed from my question, how can we make this a square number, it's clearly not one already so we're gonna split this up.

Now we need to be thinking about splitting it into 2 identical numbers.

Okay, so I've written it as 2 multiplied by 11 squared, multiplied by 2, multiplied by 11 squared, because I've noticed that I can take a factor of 2 out and I can take a factor of 11 squared then.

And then on the end I've put the 3 multiplied by 5 'cause those are the 2 prime factors that we've not yet listed.

So 2 multiply by 11 squared and we've got 2 multiplied by 11 squared.

But we've got this extra bit on the end.

Hmm, what are we gonna do about that? Well, what I'm gonna do is I'm gonna move the 3 multiplied by 5 to go with the first one to create my first integer.

Now remember, a square number is the product of 2 identical integers.

So there's my first one and I need to make this one exactly the same.

So therefore I need to introduce a factor of 3 and 5.

Now this is a square number.

You might start to think and look at the exponents here and what's special about them.

Here we've got Sam and Lucas and we've got a product of prime factors, 2 to the power of 4, multiplied by 3 cubed, multiplied by 5, multiplied by 7 squared.

So Sam says this product of prime factors is a square number because there is an exponent of 2.

So he spotted that 7 has an exponent of 2.

Lucas says this product of prime factors is not a square number because there is an exponent of 3.

Who do you agree with? So what I'd like you to do is to think about who do you agree with, do you agree with Sam or do you agree with Lucas? But also don't forget to give a reason for your answer.

So pause the video and when you are done, you can come back.

Who did you agree with? Sam or Lucas? Okay, hopefully you agreed with Lucas.

Lucas was right.

If there is an odd exponent, the number cannot be a product of 2 identical integers because if I've got 3 cubed, I can't make 2 identical integers from that.

Well done, I'm sure you got that right.

We're now going to start looking at common factors and we're going to be looking at common factors of 60 and 24.

So up until now we would've listed the factor pairs of 60.

So remember in pairs and with a system.

And then factors of 24.

So again, they're in pairs and we've listed them systematically, but actually now we know that we can write any composite number as a product of its prime factors.

We can use that to help us find common factors.

So here is 60.

60 is 2 multiplied by 2, multiplied by 3, multiplied by 5, and 24 is 2 multiplied by 2, multiplied by 2, multiplied by 3.

So here's 2, 2, they both have a common factor of 2.

Let's just check our list and make sure that that's right, it is.

Okay, we've also got 2 multiplied by 2, that's common to both primary factors, but we know that 2 multiplied by 2 is 4.

So yeah, we can check it using our list, so that's right.

We also got a 2 multiplied by 3 in both of our prime factors, so that's 6.

Yep, that's in our list.

Is there anything else that's common? We've got a 3 and a 3.

Is that in our list? Yes it is.

We've also got a 2, multiplied by 2, multiplied by 3, slightly different places, but that doesn't matter.

It is still common to both.

And 2, multiplied by 2, multiplied by 3 is 12.

And we can see that 12 is in our list.

So we want to find common factors.

We can actually use the product to prime factor form to do that.

So have we identified all of the common factors? No, we've missed one.

And actually the one we have missed is the number 1.

Why have we missed the common factor of 1? The reason we've missed that common factor is because we use prime factor form and 1, remember, is not a prime number.

A prime number must have exactly 2 factors.

So if we use this method, we must make sure that we always include the 1.

So Sam and Lucas here again.

So we've got Sam and Lucas.

"We can use these products or prime factors to identify common factors," Sam says.

Hmm, so Lucas is quite interested to know how you can do that.

"We can look for what is common in both." So just like we did with the previous example, but this time we're actually got, we've got it in exponent form.

"So 2 is a common factor?" "Yep, yeah, but there are lots more." So let's take a look and see if we can find them all.

So we're gonna list all the common factors of these 2 products and we'll do this systematically, 'cause remember we like to do things systematically so that we know that we've probably not missed any, so we're gonna do it systematically.

So let's look first, is 2 common to both? Yes, it is, so 2 is a common factor.

Right, now let's think about 2 squared.

Is 2 squared common to both? Yes, because remember 2 cubed is actually 2 squared multiplied by 2.

So 2 squared is also a common factor.

What about 2 cubed? Is 2 cubed common to both? No, the first product only has 2 squared, only has a repeated multiplication of 2 twice.

So therefore we can't make a repeated multiplication of 2 3 times.

Now we'll move on to the 3s.

So 3, yes, that's in both.

3 squared, yep, again, that's in both.

Is 3 cube common to both? Yes, it is because 3 to the power of 5, remember it's just 3 cubed multiplied by 3 squared.

So 3 cubed is common both.

So we'll put that in our list.

Why don't we need to check 3 to the power of 4 and 3 to the power of 5? The reason we don't need to check those is because the highest common factor is 3 cubed.

Is 7 common to both? Yes, also remember, any product of those is also going to be a common factor of both of those products.

So there's an example there, 2 squared multiplied by 3 cubed multiplied by 7.

So any product of those is going to also be a common factor.

Let's check in now and make sure that we are ready to move on to the next task.

So true or false, 3 squared multiplied by 5 squared is a common factor with 3 to the power of 4 multiplied by 5, multiplied by 11 and 2, multiplied by 3 squared, multiplied by 5 squared? So as always, I don't want you to just choose true or false.

I want you also to pick the justification, which is the correct justification for that? So pause the video and when you are done, you can come back and see how you've got on.

Okay, well done, so hopefully you decided it was false and the correct justification was b.

The first one only has a factor of 5, not 5 squared.

Shouldn't I be ready to tackle task B? So this is quite a challenging task.

So remember if you need to, you could go back and you could watch any part of the video again, just to make sure that you are happy to do this task.

Using prime numbers, make each of the following products satisfy the condition on the right.

So I've given you some product to prime factors on the left, and what I want you to do is to make them satisfy what's written on the right.

So for example, for a, introduce another factor, and then it needs to be an even number.

And remember here again, we are not popping these into our calculators, we are using the information that we've got already and our knowledge of prime numbers.

So you can pause the video now, and then when you are done, you can come back.

Now we can move on to question number 2.

Here is a number written as a product of its prime factors.

Right, again, I don't want you to work it out, so I don't want you to work it out.

And then what I want you to do is to decide whether you think the following statements are true or false.

And remember, I'm not happy with just true or false, I want to know, you need to show me that you understand why by justifying your answers.

Good luck with this, it is really challenging task, but you have everything you need to be able to be successful at it.

So pause the video and as always come back when you're ready.

And here we've got a third question in this task B.

So this time I want you to list all of the common factors of 2 to power of 4, multiplied by 3, multiplied by 7 squared, and 2 cubed multiplied by 3 cubed, multiplied by 7.

So remember, we wouldn't want to be using a systematic way of listing these in factor pairs.

We would be here for a very long time.

So we are looking for what is common to both of those products.

Good luck with that, pause the video and then come back when you're ready.

Well done, let's now check our answers.

So number one, so A, remember if it says e.

g.

, that means it's just an example, you may have something different, but I've gone probably for the easiest way.

Okay, so we've got first one, multiply by 2, second one multiplied by 3, c, multiplied by 3 squared, d, multiplied by 2, e, multiplied by 5, multiplied by 13, and f, multiplied by 3, multiplied by 11 squared.

Now let's check our answers to question number 2.

And remember, I'm not going be very impressed if all you've written is true or false, but I'm sure you haven't.

We must have those justifications to make sure that we've truly understand what we're doing.

So a is true because 2 is a factor, b, true since 2 and 5 are both factors, c, true the prime factors of 100 are 2 squared and 5 squared, d, true, the prime factors of 35 are 5 and 7, e, true, the prime factors of 50 are 2 and 5 squared, and f, true, it can be written as 2 squared multiplied by 5, multiplied by 7 all squared.

I bet you didn't think they were all gonna be true, did you? And now our answers to question number 3.

So remember here I was asking you to list all of the common factors.

I'm not gonna read all of these out.

What I'm gonna say to you is to pause the video and then check your answers.

Let's finish up by summarising what we've done in today's lesson.

You've done fantastically well.

Products of prime factors can be evaluated to give the integer.

So remember, we can use our calculator, we can input the product of prime factors and then we can evaluate that to give the integer, remembering to use your calculator.

And when you're doing that, use the exponent button.

This is because it's more efficient and also you are less likely to make an error.

We then looked at products of prime factors and we know now that we can use them to identify common factors of numbers.

So we know that we can find what's common in both.

And products of prime factors can be used to determine properties of integers.

So remember we looked at making things multiples of 10 or checking if it was a square number and how to make it into a square number, et cetera.

I'm really pleased that you decided to join me for this lesson today.

Well done, you've done fantastically.