Lesson video

Hi there, 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.

The title of today's lesson is Highest Common Factor and it's in the unit properties of number.

By the end of today's lesson, you'll be able to use prime factorization of two or more positive integers to efficiently identify the highest common factor.

So key word for today's lesson, key words, is prime factorization.

This is a method to find the prime factors of a given integer.

The highest common factor of two or more intergers, which can be divided by all other possible common factors.

And HCF, HCF is an abbreviation for the highest common factor.

It's quite a mouthful to say, so we can abbreviate it to HCF.

So if you hear me use that term or you see it on the screen, then you'll know it basically stands for highest common factor.

We'll also be using the term exponent, and the exponent is a number positioned above and to the right of a base value, and it indicates repeated multiplication.

You should be aware of that and familiar with that.

So today's lesson, we are going to spit into three separate learning cycles.

We're going to concentrate first on common factors using prime factorization.

We'll then move on to highest common factor using prime factorization, and then we'll move on and look at what happens if we have more than two numbers.

So we're going to start with common factors using prime factorization.

So here, we've got two integers written as a product to their prime factors, what is the same and what is different about these two products of prime factors? I'd like you now to pause the video and write down what you think is the same about them and what is different.

Well done, so let's have a look and see.

You may have written some of these, so you could have said something like, they are both products, they both use the intergers two, three, and five, they have a common factor of five cubed, they have a common factor of two squared multiplied by five cubed.

The exponents of the twos are different, the exponents of the threes are different.

You may have said some things like that, you may have said something different.

We can use prime factorization of integers to identify common factors, and it's useful to do this systematically and beginning with with the lowest exponent of each prime factor.

So here, we would start with two squared.

So I've highlighted the twos in each of our product prime factors, two squared is common to both products, because remember, two cubed is actually two squared multiplied by two.

As two squared is common to both products, then two must also be a common factor.

Next, we'll take a look at the prime factor three.

So I've highlighted those in each of our product prime factors.

Three is common to both products.

And now let's look at five.

So we're starting with the lowest exponent, that's five cubed.

So if five cubed is a common factor, then also five and five squared must also be common factors.

Here is a list of all of the common factors we have identified, so two, two squared, three, five, five squared, and five cubed.

And Jacob says we found all of the common factors of two squared multiplied by three multiplied by five cubed, and two cubed multiplied by three to the power of four multiplied by five to the power of five.

Do you agree with Jacob? Jacob is not right, because if two and three are both factors, two multiplied by three must also be a common factor.

We need to include products of any combination of the factors we have identified.

So if, for example, two squared multiplied by three multiplied by five cubed would also be a common factor, but remember, that's not the only one, any combination of the common factors we've found Andeep, Laura, and Jun are working out common factors of three to the power of four multiplied by five squared multiplied by 11 squared, and three squared multiplied by five multiplied by 11 to the power of four.

Who has not correctly identified a common factor? So let's have a look at what they've all said.

So Andeep says a common factor is three squared multiplied by five multiplied by 11.

Laura says a common factor is three squared multiplied by 11 squared.

and Jun says a common factor is three squared multiplied by five squared multiplied by 11 to the power of four.

You need to say who has not correctly identified the common factor.

Pause the video and have a think about it, and then come back when you're ready.

Okay, so it was Jun.

Jun has not identified a correct common factor.

The highest common factor of 11 squared and 11 to the power of four is 11 squared.

So here, he's got 11 to the power of four, that is not common to 11 squared and 11 to the power of four.

Okay, now we should be ready to have a go at our first task.

So in this task, we are going to be identifying the incorrect common factor of the two products given in the first column and there's just one in each row.

So you look at the first column, so for example, two to the power of four multiplied by three squared multiplied by 11, and two squared multiplied by three to the power of five multiplied by 11 squared.

You'll then need to decide which one is an incorrect common factor of those two products given.

So pause the video and then come back when you're ready to check in with those answers.

Brilliant, well done.

So now we're just going to check those answers.

So for the first one, we should have two to the power of four multiplied by three to the power of five multiplied by the 11 squared.

For the second one, three cubed multiplied by seven multiplied by 13 squared.

The third one, two to the power of four multiplied by three to the power of four multiplied by five squared multiplied by 19 squared.

The fourth one, three to the power of four multiplied by seven squared multiplied by 11 to the power of seven.

And then the final one, it was two to the power of four multiplied by five squared multiplied by seven.

Well done if you've got all of those right.

Fantastic.

We're now gonna move on to our second learning cycle.

So we're now going to be looking in particular for the highest common factor.

So we've just looked at common factors.

We're gonna move on and look specifically for the highest.

So we are going to find the highest common factor of 24 and 40.

So previously, we would've done that by listing all of the factors of 24 and 40.

The highest common factor is the highest number that appears in both lists.

So in this case, it would be eight.

Eight is the highest number that appears in both lists.

So the highest common factor of 24 and 40 is eight.

Now I want you to have a think now, do you think that would be an efficient method if we wanted to find the HCF of 240 and 400, for example? So much, much bigger numbers.

Do you think that would be efficient? No, this would not be an efficient method.

Luckily, we can find the highest common factor of two numbers using prime factorization and a Venn diagram.

So we've got here 240 written as a product of its prime factors, and 200 written as a product of its prime factors.

So we're now gonna put those into our Venn diagram.

So in our Venn diagram, we're gonna have the prime factors of 240 in the left set and 200 in the right one.

And then remember, if it's common to both, we are gonna put that in the middle.

So let's go through these.

So I can see here clearly that two appears in both of them, so it's going to go in the intersection, the middle part of the Venn diagram.

Let's now have a look at the next one, so the next prime factor, okay, is two, and we can see again that that appears in both, so we'll pop that in the intersection.

And then there is another two, another third two that is common to both of my products, so that's gonna go in the intersection.

Now, using this system, I'm now gonna go straight onto looking at the next one from the left hand number, from 240, which is two, but there is not another two as a factor in 200, so that is just going to go in the left side.

Let's now move on to the next one, which is three.

Three, again, there is no factor of three in 200, so that's gonna go in the left.

And then five, this time it is common to both of our products, so it's going to go in the intersection, the middle, and then we've got the five is the final one to go on the right hand side of the diagram.

So we've put all of the common factors into the intersection of the Venn diagram.

Therefore, the product of these will give us that highest common factor, the HCF.

So the highest common factor 240 and 200 is the product of the common factors, two multiplied by two multiplied by two multiplied by five, and that gives us 40.

We're now gonna have a go at doing a question, so we're going to do the one on the left hand side together, and then you are going to independently have a go at the one on the right hand side.

So to help us with this so that we don't have to write the numbers of products of primes, I've given them to you.

So 840 and 2,772, so let's go through just like we did with the previous one.

So start with the first prime factor, it's in both, so we're gonna pop that in the intersection.

The next prime factor is two, again, it's in both.

It's going in the intersection.

Then there's a two, it's in 840, but not in 2,772, so that's gonna go in the left.

Then the next is three.

We can see here again that that is common to both.

So we're gonna put that in the intersection.

Okay, there is another three, so we're now gonna move on to three, so three, this time it was only in the bottom number, so that's gonna go in the right hand side.

Next prime number is five, so five is only in 840, and then the next prime number is seven.

That's common to both, so therefore, we're gonna put it in the intersection.

And then finally, the 11 is just going to the right hand side.

So remember, we've put all of the common factors in the intersection of the Venn diagram, and we now need to find the product of those.

So the HCF of 840 and 2,772 is the product of two multiplied by two multiplied by three multiplied by seven, and we can use calculator to work that out.

The answer is 84.

You should now be ready to have a go at one of these independently, but don't worry if you're not, you could always go back and watch the two examples again.

So this is a question that I would like you to have a go at, find the highest common factor of 1,260 and 1,638.

I've given you there the two product of prime factors, and then this is your Venn diagram.

So what I'd like you to do is have a go at this question, so pause the video and then come back when you are ready.

Remember, it's okay to use a calculator for that final step for finding that final product.

Great work.

Let's have a look then and see if you've got it right, which I'm sure you have.

So we should have our Venn diagram filled in, and then we remember we are finding the product of the common factors, and the common factors are in the intersection, middle part of our Venn diagram.

So the correct answer was 126.

Well done if you got that right.

If you didn't, maybe just have a look and see was it the Venn diagram or did you make error when you calculated the product? So here we have Izzy.

Izzy is using the Venn diagram to find the highest common factor of 150 and 60, but unfortunately, she's made a mistake.

What mistake has Izzy made? Pause the video and have a look and see if you can find Izzy's mistake.

She's put the common prime factors into the intersection twice.

So notice there was a two common in both 150 and 60, but it should only be in the intersection of the Venn diagram once.

The same with three and the same with five.

How could Izzy find out what numbers are represented in her diagram? A really good way of her checking to make sure she hadn't made any errors.

So to find that, we would just find the product of each set, so we multiplied together, and if we did that, Izzy would've been able to spot that actually, the left hand set didn't give her 150 and the right hand set didn't give 16, so spotted that error.

What is the same and what is different about these Venn diagrams? Pause the video and write down what you think is the same and what you think is different about these two diagrams, They're the same Venn diagram, the only difference is the bottom one has written five multiplied by five in its exponent form.

So we can see that the only difference is that in the intersection, the top one says two, three, five and five, and on the bottom one we've got two, three, and five squared.

So it is possible to use the exponent form when writing numbers into the Venn diagrams. We;re now ready to do task B.

So for task B, you are going to have a go at finding the highest common factor of those pairs of numbers.

So remember, you are going to put the common factors in the middle, don't put them in twice unless they are common twice, and then you are going to find the product of the intersection.

So pause the video and come back when you're done.

Question two for this task, so this one is a little bit more challenging.

I've given it to you the other way round, so rather than getting you to put in the numbers in the Venn diagram, you can have a go at this one.

What is the missing prime factor? And what are the two numbers? So we're told that the highest coin factor of two numbers represented in the Venn diagram is 210.

So there is a missing prime factor, and then also, what 210 is the highest common factor of which two numbers? Again, pause the video and come back when you're ready.

Amazing job.

Right, let's check those answers.

So 1A, the answer is 28, B, the answer is 98, and C, the answer was 504.

Question number two, so with this question, remember, the highest common factor is the product of the numbers in the intersection.

So here we've got two multiplied by three multiplied by seven.

To make that product become 210, we need to introduce a factor of five.

And part B, the left number is the product of all of the prime factors in the left set.

Remember to include that five.

So 630, and the right number again is the product of all of the numbers in that set and the introduction of that five, which is 1,050.

Well done if you've got all of those right.

We're now gonna move on to our final learning cycle in this lesson, which is highest common factor of more than two numbers.

Now, if you can do the highest common factor of two numbers, you can definitely do the highest common factor of more than two numbers.

So let's have a look and see what that looks like.

Alex and Izzy drew a Venn diagram to find the highest one factor of 84, 180, and 63.

Who has drawn the correct Venn diagram? So Alex draws this one and Izzy draws this one.

Who do you think has drawn the correct Venn diagram? Okay, it's Izzy.

Izzy has drawn the correct diagram.

The problem with Alex's one is what happens if there is a factor that is common to all three numbers? Then there is no intersection of all three sets.

So we've drawn our Venn diagram just like Izzy's.

We've got the numbers 84, 180, and 63, and we're going to put the prime factors into the Venn diagram.

So starting with two, so two squared appears in 84 and 180, so that goes in the intersection of just those two.

Right, let's move on to three then.

So three, we've got three for 84, we've got three squared for 180, and we've got three squared for 63.

Is there anything that's common to all three of them? Yes, there's a three that's common to all three of them, and then we are looking to see if there's anything else that's common.

So remember, three squared is three multiplied by three.

We've already put a three into the set for 180 and 63, so we need a second three, and it's common to 180 and 63, so it needs to go in the intersection of those two numbers.

Let's move now on to the next prime number.

That's five.

We notice here that five only appears as a factor in 180, so it's going to go just in 180.

And then seven, seven appears in 84 and 63, so it needs to go in the intersection of those two sets.

So the highest common factor of 84, 180, and 63 is three.

Remember, it's the highest common factor of all three of those, so we need the intersection of all three numbers.

So it's three.

Let's have a little check for understanding now.

Which prime factor is in the wrong place? And where should it go? So I've completed this Venn diagram, but unfortunately, I've made an error.

What I'd like you to do is to pause the video and work out what my error is and then correct that error.

So put it right.

So the one that was in the wrong place was a two.

Okay, so the two is actually, we've got the two in the intersection of all three, but then 252 is two squared, and so therefore we've already put one two into the intersection, the two should have been just a prime factor of 252.

So we think about that, if the two had been where it was originally, then it would've needed to appear in both the numbers 90 and 252.

Well done if you got that right.

Onto our final task for today's lesson.

Now, this is extremely challenging.

It's really challenging, but I know that you've got all of the skills to be successful.

So you might need to pause the video and then just go back and rewatch some things maybe.

Everything we've done today has given you the skills to be able to complete this task.

So you need to fill in the blanks.

Pause the video, and then come back when you are ready to check your answers.

Wow, that was quick.

Let's have a check then.

So hopefully we've spotted that 360 was the missing number at the top, and then there was a five missing from that product of prime factors.

There was a three missing from prime factors of 84.

The third number was 9,900.

The missing exponent of five was two, then missing in the Venn diagram was a three in the intersection of all three numbers.

Highest common factor of 360, 84, and 9,900 is 12.

Absolutely superb if you've got that right, and like I said, it was a really challenging problem, so well done.

Now let's just summarise what we've done this lesson.

Common factors can be found using prime factorization.

So previous to today's lesson, we probably would've just listed the factors and then we would've looked for the highest one in each, but that becomes inefficient when we're looking at larger numbers.

The highest common factor of numbers can be found using prime factorization and maybe a Venn diagram.

Some of you might be able to do it without the Venn diagram and just visualise that Venn diagram, which is absolutely fine.

So the example there is how we found the highest common factor of 450 and 1,500.

You've done absolutely superbly today.

I've really enjoyed working through the highest common factor with you.