Lesson video

Hi, there.

My name is Ms. Lambel.

You've made a really good decision to decide to join me today to do some maths.

Come on, then, let's get started.

Welcome to today's lesson.

The title of today's lesson is "Checking and Securing Understanding of Prime Factors," and that is within the unit Arithmetic Procedures and Index Laws.

By the end of this lesson, you'll be able to write any positive integer uniquely as a product of its prime factors.

Keywords that we're going to be using in today's lesson, let's just recap those to help us with our understanding of what we're going to be doing.

Expressing a number as a product of primes means writing it uniquely as a product of its factors that are prime numbers.

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

Prime factorization is a method to find the prime factors of an integer.

So we will be using prime factorization to write number as a product of its prime factors.

Today's lesson is split into two learning cycles.

In the first one, we are going to write numbers as a product of their prime factors, and in the second one we will use that to help us answer some of the questions.

Let's get going with that first one, writing numbers as product of prime factors.

We are going to write 1,800 as a product of its prime factors.

Remember the word product means a multiplication.

1,800 could be written as the product of 18 multiplied by 100.

We then repeat that process.

18 could be written as the product of 2 multiplied by 3 squared.

3 squared is 9, so let's just check.

2 multiplied by 9 is 18.

Here I've decided to save myself a little bit of time by writing 9 as 3 squared because I know that 9 is 3 squared.

I now need to write 100 as a product of two numbers and I've chosen to do 10 and 10.

We repeat that.

2 and 3 are prime numbers, so they are already in prime number form, so we're going to repeat the process with the 10 and the 10.

What two numbers have a product of 10? Well, that's 2 and 5, and obviously for the other 10, that's also 2 and 5.

Now, we can collect together all of our prime factors.

Our answer is 2 cubed multiplied by 3 squared, multiplied by 5 squared.

Where does that 2 cubed come from? Well, we've got a repeated multiplication of 2, three times.

3 squared, we only had a 3 squared in our product of primes.

And then 5 squared because we had a repeated multiplication of 5 twice.

1,800 is 2 cubed multiplied by 3 squared, multiplied by 5 squared when it is written as a product of its prime factors.

Let's give it another one a go.

We're going to write this time 1,008 as a product of its prime factors.

1,008 is 4 multiplied by 252.

We can then break down the 4.

We know that 4 is 2 squared.

Now, let's break down 252 as a product of two factors.

And I've chosen to do 2 squared, which is 4 and 63.

The 2 squared and the 2 squared are gonna remain the same 'cause they're already prime numbers.

And we are then going to write 63 as a product of two factors, and I've chosen 3 squared and 7.

3 squared is 9 and I know that 9 multiplied by 7 is 63.

The answer then is 2 to the power of 4, multiplied by 3 squared multiplied by 7.

Where does that 2 to the power of 4 come from? Well here, we've got a repeated multiplication of 2 twice, multiplied by a repeated multiplication of 2 another two times, so in total, that is repeated multiplication of 2 four times.

And then we've got the 3 squared and we've got the 7.

1,008 is 2 to the power of 4 multiplied by 3 squared multiplied by 7 when it's written as a product of its prime factors.

One more together and then I know you'll be ready to have a go at one independently.

This time, we're going to write 396 as a product of its prime factors.

Let's break it down.

396 is a product of 4 and 99.

4 is 2 squared.

99 is 3 squared multiplied by 11.

2, 3 and 11 or all prime numbers, so have written it as a product of its prime factors.

396 is 2 squared multiplied by 3 squared multiplied by 11 when it's written as a product of its prime factors.

Now over to you.

I'd like you, please, to write 1,400 as a product of its prime factors.

Pause the video, and then when you've got your answer, come back.

Well done.

Let's check, make sure we've got the same answer.

1,400, I decided to break it down into the factors of 4 and 350.

I know that 4 is 2 squared, and I know that 350 is 35 multiplied by 10.

Then I need to break my 35 down into a factor pair, 5 and 7, and the same with 10, which is 2 and 5.

And then we're going to tidy this up by collecting together the same prime factors.

So we end up with 2 cubed multiply by 5 squared, multiply by 7, so 1,400 is 2 cubed multiplied by 5 squared, multiplied by 7 when it's written as a product of its prime factors.

Now, you will have exactly the same answer as me, even if you chose to use different factors throughout the method.

Remember each composite number has a unique product of prime factors.

It doesn't matter how you choose to break it down, you'll end up with 2 cubed multiplied by 5 squared, multiplied by 7.

Here, we've got Sam and we've got Izzy and they are working on this problem.

It's a who am I problem.

I am a multiple of 10.

I am the product of three unique prime numbers.

None of my prime factors are two digits.

There are no exponents in my product of prime factors.

I am greater than 6 squared.

Sam says, "Okay, so using the third clue, I know that it must be the product of three of 2, 3, 5 and 7." Yeah, that's right isn't it? Because the third clue says none of my prime factors are two digits, and the next prime number after 7 is 11, and that's two digits.

Izzy says, "And we know that two of them are 2 and 5." And Sam's reply is, "How do you know that?" Do you know how Izzy knows that two of them have to be 2 and 5? Well, let's see what Izzy's got to say.

Because it is a multiple of 10, the first clue said that number is a multiple of 10 and the prime factors of 10 are 2 and 5.

Brilliant, well spotted, Izzy.

Sam says, "Of course, so it must be the product of 2, 5 and 3, or 2, 5 and 7.

The product of 2, 3 and 5 is 30, which is less than 6 squared, so if we look at the final clue, it says I'm greater than 6 squared.

Sam says, "Then it must be the product of 2, 5 and 7, then." Izzy says, "Yes, the number is 70." So using those clues, when I first looked at that, can we work out what the number is? But actually, Sam and Izzy have used their knowledge of prime numbers and prime factors to work out what the number was and the number was 70.

Now you're ready to have a go at task A.

In question number one, you are going to write each of these six numbers as a product of its prime factors.

You are not going to use a calculator, and try to be as efficient as you can.

Pause the video, and then when you're ready, come back and we will have a look at question number two.

Good luck.

Great work.

Question number 2A is a who am I question.

I am the product of four different prime numbers.

I am even.

I am a multiple of 5, but not a multiple of 7.

My largest prime factor is nine greater than my smallest prime factor.

I am less than 500.

Who am I? Pause the video.

Use your detective skills and prime factor skills to work out which number this is.

And on to B, another who am I.

I am a product of five prime numbers.

I am a multiple of 10.

My highest exponent is 2.

This is not the exponent of the smallest or largest prime factor.

My largest prime factor is 7.

I am less than 1,000.

Who am I? Pause the video, use your skills, come back when you know which number it is.

And part C.

I am a square number.

I am a multiple of 100.

The difference between my highest and lowest prime factors is 3.

I am less than 10,000.

My highest exponent is 4.

Find both possible answers.

So, this time there is not just one unique answer, there are two answers and I'd like you to find both of them, please.

Pause the video, come back when you've worked out what they are.

Great detective work.

Now, let's move on to looking at the answers.

Question 1A, 84 is 2 squared multiplied by 3, multiplied by 7.

B, 180 is 2 squared multiplied by 3 squared multiplied by 5.

C, 588 is 2 squared multiplied by 3 multiplied by 7 squared.

D, 792 is 2 cubed multiplied by 3 squared multiplied by 11.

E, 5,096 is 2 cubed multiplied by 7 squared multiplied by 13.

And F, 39,325 is 5 squared multiplied by 11 squared, multiplied by 13.

Remember, you may have those prime factors in a different order.

I just always like to write them in numerical order.

And who am I? A, I am 330.

B, I am 630.

C, I am either 3,600 or 8,100.

How did you get on with those number detective problems? Fantastic, well done.

Now, we can move on to the second learning cycle.

We're gonna use the prime factor form now.

Sam and Izzy's teacher has asked them to write down a factor of 252 that is between 40 and 50.

Izzy says, "It's gonna take ages to write down all of the factor pairs of 252." Sam says, "I think we can use the product of prime factors." Do you think Sam's right? Remember, we can use our calculator to help to find the product of prime factors.

So, this is the calculator that I'm using and I'm gonna show you how to do it on this calculator.

Your calculator may be slightly different now, but if you've got calculator similar to this one, this is probably the way that you're going write a number as a product of these prime factors.

First, we need to ensure that calculator's highlighted and then we press the EXE button.

We're gonna check the prime factor decomposition of 252.

We type in 252 and then press EXE.

You will then have this on your screen.

We then select format, so we click format button.

You should have this on your screen.

We then scroll down until prime factor is highlighted, and then we press EXE again.

And this will reveal the number as a product of its prime factors, 2 squared multiplied by 3 squared multiplied by 7.

Sam and Izzy's teacher has asked them to write down a factor of 252 that is between 40 and 50, so that's what they were working on, and Sam suggested that writing 252 as a product of its prime factors may help us answer this question.

So 252 equals 2 squared multiplied by 3 squared, multiplied by 7, and Izzy wants to know how that's gonna help us.

Izzy's still not sure how that's gonna help.

And Sam's response is, "A product of any combination of these will also be a factor of 252." Izzy says, "Of course." She's gotten on now, she's worked out that, yes, actually, a combination of those and finding the product of those will also be a factor of 252.

The product of 2, 3 and 7 is 42, so that's our answer.

Yes, Sam's managed to find a factor of 252 that is between 40 and 50.

She then asks, "Which of the following is not a factor of 252?" So we've got 9, 12, 15, 36, 49 and 63.

We're gonna use the prime factors.

So here is 252 written as a product of its prime factors.

Let's start with 9.

Well, 9 is 3 squared and we can see clearly in that product of prime factors there's a 3 squared, so yes.

12, as a product of its prime factors, is 2 squared multiplied by 3, and we can see that that is in our prime factor decomposition that we've got for 252.

15, 15 written as a product of its primes is 3 multiplied by 5.

If we check, there is no 5.

This is not going to be a factor because there is no factor of 5 in the product of prime factors of 252.

36 is 2 squared multiplied by 3 squared.

That does appear.

49 is 7 squared, so this one is not going to be a factor.

Only one prime factor of 7 is in the product to prime factors.

And finally, 63 is 3 squared multiplied by 7, and we can clearly again see that that is a factor because it appears in the product of prime factors.

I'd like you to use that 1,575 is equal to 3 squared multiplied by 5 squared, multiplied by 7 to identify which of the following are factors of 1,575.

Pause the video, make sure you're using that method that we've just been through.

When you're ready to come back, we'll check those answers.

Great, well done.

Let's check those answers, so A is correct.

And you should have identified that 75 is 3 multiplied by 5 squared and we can see that that appears.

125 was no, 125 is 5 cubed.

And if we look at the product prime factors for 1,575, we can clearly see that it's only got 5 squared.

175 is 5 squared multiplied by 7 and that appears.

And 225 is 15 squared, which I know is 3 squared multiplied by 5 squared, and that could be written as 3 multiplied by 5 squared and we can clearly see that that is also in the product prime factors of 1,575.

360 equals 2 cubed multiplied by 3 squared multiplied by 5.

We can use this to write other numbers as a product of their prime factors without repeating the process from the beginning.

So for example, 1,800 is 360 multiplied by 5.

We already know that 360 as a product of its prime factors is 2 cubed multiplied by 3 squared multiplied by 5.

We then just need to multiply this by 5, and then we are going to collect together our prime factors.

2 cubed multiplied by 3 squared, multiplied by 5, multiplied by 5, which is 5 squared.

1,440 is 360 multiplied by 4.

We know that 360 is 2 cubed multiplied by 3 squared multiplied by 5, and we're multiplying that by 4, which is 2 squared.

This is 2 to the power of 5 multiplied by 3 squared multiplied by 5.

One more and then you can have a go at one independently.

Use the product prime factors to write 3,564 as a product of its prime factors.

3,564 is 396 multiplied by 9.

396 is 2 squared multiplied by 3 squared multiplied by 11, and nine is 3 squared.

That's where those come from.

And then finally, we can just collect together our repeated multiplication of the prime factors, so we end up with 2 squared, 3 squared multiplied by 3 squared is 3 to power 4 multiplied by 11.

And that's where the 3 to the power of 4 came from.

It came from the product of 3 squared and 3 squared.

Now your turn.

Pause the video, give this one a go, and come back when you're ready.

Well done.

1,584 is 396 multiplied by 4.

We know that 396 as prime factors 'cause we're told that in question.

4 is 2 squared, so 396 we were given in the question, 4 is 2 squared and 2 to the power 4 multiplied by 3 squared multiplied by 11, 2 to the power 4 comes from the product of 2 squared and 2 squared, so our answer is 2 to the power 4 multiplied by 3 squared multiplied by 11.

Now let's take a look at this one.

Given that 220 equals 2 squared multiplied by 5 multiplied by 11, and that y equals 180 multiplied by 220, write y as a product of its prime factors.

Now what we could do is we could work out 180 multiplied by 220, and then find the prime factors of that.

That's gonna give us a really big number and we already know what 220 is as a product of its prime factors.

So firstly, we're going to write 180 as a product of its prime factors.

180 is 2 multiplied by 90, which is 2 multiplied by 3 squared multiplied by 10, so 90 is 3 squared multiplied by 10, which is 2 multiplied by 3 squared multiplied by 2, multiplied by 5.

And then we know that 2 multiplied by 2 is 2 squared and so 180 is 2 squared multiplied by 3 squared multiplied by 5.

Y was 180 multiplied by 220.

180 we've just worked out, well, that is a product with prime factors.

It's 2 squared multiplied by 3 squared multiplied by 5 and we're multiplying that by 220 and we were given that as a product of its prime factors in the question which was 2 squared multiplied by 5, multiplied by 11.

2 squared multiplied by 2 squared is 2 to power 4.

And then we've got 3 squared, then we've got 5 multiplied by 5, which is 5 squared and 11.

So the answer is 2 to the power of 4 multiplied by 3 squared, multiplied by 5 squared, multiplied by 11.

Now we'll do one more of those together.

We need to write y as a product of its prime factors.

180, we've just worked this out.

Now we're gonna multiply it by 250.

So we've got 180 is 2 squared multiplied by 3 squared multiplied by 5.

And we're given in the question that 250 is 2 multiplied by 5 cubed.

2 squared multiplied by two is 2 cubed, and then 3 squared, and 5 multiplied by 5 cubed is 5 to the power 4 because there would be a repeat in multiplication of 5 four times.

Now your turn.

Pause the video, and then when you've got your answer, come back.

Super work, let's check.

140, I started with 10 and 14.

Remember, you may have started with something different.

That's absolutely fine.

Which is 2 multiplied by 5, multiplied by 2, multiplied by 7, which is 2 squared multiplied by 5, multiplied by 7, and then we're gonna multiply that by 250, which is the 2 multiplied by 5 cubed.

2 squared multiplied by 2 is 2 cubed.

5 multiplied by 5 cubed is 5 to the power of 4, multiplied by 7.

Now let's take a look at this one.

Now don't let the fact that there is an algebraic exponent for the 5 confuse you.

We're gonna do exactly the same thing as we've just done.

We're going to write 9y as a product of its prime factors.

We know that y as a product of its prime factors is 2 multiplied by 3 to the power of 4, multiplied by 5 to the power of A.

So if y equals that, then 9y is going to be 9 lots of that, so nine multiplied by that.

But we know that 9 is 3 squared, and then we can just tidy this up.

So we've got 9 is 3 squared.

We're gonna tidy that up, 3 to the power of 6.

Where's that come from? Well here, I've got a repeated multiplication of 3 twice, a repeated multiplication of 3 four times, which means a repeated multiplication of 3 six times in total.

9y written as a product of its prime factors is 2 multiplied by 3 to the power of 6, multiplied by 5 to the power of A.

It doesn't matter that we didn't know what the exact exponent for the 5 was.

Y equals 2 multiplied by 3 to power 4 multiplied by 5 to power of A.

Write 8y as a product of its prime factors.

I'd like you to pause the video, decide whose answer is correct, or whose workings are correct, and explain the mistake that the other person has made.

Pause the video, and then when you've got your answer, come back, All done, who got it right? It was Izzy.

Izzy got it right.

What mistake does Sam make? Sam has missed a factor of 2, so if we look at the middle line of Sam's working, we can see he's got 2 cubed multiplied by 2, which is actually 2 to the power of 4.

He's put 2 cubed.

That's Sam's mistake.

Now on to task B, write down at least 5 different factors of the following numbers using the product to prime factors.

So I'd like you to write down at least 5 different factors using the product to prime factors that you've been given.

Pause the video, and then when you've got your answers, you can come back and I'll reveal question number two.

Question two, use the fact that 144 equals 2 to the power of 4 multiplied by 3 squared to write the following as a product of their prime factors.

Pause the video.

Again, when you're ready, come back.

Remember, no calculators.

Good luck and I'll be here awaiting when you get back.

Question number three.

If y equals 2, multiplied by 3 to the power 4, multiplied by 5 to the power of A, multiplied by 7 to the power of B, I'd like you to write 2y, 3y, 6y, and 36y as a product of their prime factors.

You can pause the video now and come back when you're ready.

Absolutely superb work, well done.

Let's check our answers.

1A, you can have any of these.

I'm not going to read them out.

You can pause the video and then you can check yours, but remember, these are just some examples.

There are other factors that would be okay.

Question number two, A is 2 to the power of 5 multiplied by 3 squared.

B, 2 to the power of 4, multiplied by 3 cubed, multiplied by 7.

C, 2 to the power of 6 multiplied by 3 cubed.

D, 2 to the power of 4 multiplied by 3 squared, multiplied by 5.

E, 2 to the power of 7 multiplied by 3 cubed, multiplied by 5.

And finally, in question number two, F, 2 to power 6 multiplied by 3 squared, multiplied by 11 squared.

And if you need to look, 'cause if you've made any mistakes, I've tried to show you there where those answers have come from.

And then onto question number three, A, 2 squared multiplied by 3 to the power of 4, multiplied by 5 to the power of A, multiplied by 7 to the power of B.

B is 2 multiplied by 3 to the power of 5, multiplied by 5 to the power of A, multiplied by 7 to the power of B.

C, 2 squared multiplied by 3 to the power of 5, multiplied by 5 to the power of A, multiplied by 7 to the power of B.

D is 2 cubed multiplied by 3 to the power of 6, multiplied by 5 to the power of A, multiplied by 7 to the power of B.

How did you get on with those? Super, brilliant, well done.

Now we can summarise the learning from today's lesson.

Numbers greater than one are either composite or prime.

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

So for example, 52 is equal to 2 multiplied by 26, but 26 can be written as 2 multiplied by 13.

Products of prime factors can be written more efficiently using index notation, so write in 2 multiplied by 2 as 2 squared.

You've worked really, really hard today, well done.

I look forward to seeing you again soon.

Goodbye, take care.