Lesson video

Hi there, and welcome to another math lesson with me, Dr.

Saada.

In today's lesson, we would be looking at the highest common factor and prime factors.

Don't worry if you don't know what that is yet, you will by the end of today's lesson.

All you need for today's lesson is a pen and a paper, so if you do not have these handy, please pause the video, go grab them, and when you're ready, we can make a start.

Your first task for today's lesson is to write 24 and 36 as a product of prime factors.

What factors do they have in common? If you're confident about this, please pause the video now and have a go at it.

If not, don't worry, I'll give you a hint in three, two, and in one.

Okay, so your hint is to start this factor tree diagram, something that you have done in your previous lessons.

So write now 24.

Can you do two branches? Can you think about the two numbers that multiply to give you 24? So 2 is one of those numbers, what's the other one? 2 multiplied by? Really good.

Why did I circle 2? Excellent.

I circled 2 because it's already, it's a prime number.

So now, you write in the next number and then you keep doing that, writing it as a product of its prime factors until you have 24 written completely as a product of its prime factors.

With this hint, you should be able to have a go at this question.

The try this task should take you about five minutes to complete.

Please pause the video and complete it to the best of your ability.

Resume once you're finished.

Welcome back.

How did you go on with this task? Did you manage to finish it? Really good.

I guess the first one, 24, I can write it as 2 multiplied by 12, 2 is prime so I circled it, 12 is not, so I can write the 12 as 3 multiplied by 4.

The 3 is a prime number, I can circle it, the 4 is not, and the 4 can be written as 2 multiplied by 2, where both obviously are prime numbers.

And now, I can write a 24 is equal to 2 multiplied by 2 multiplied by 2 multiplied by 3, or I can write it down as 24 is equal to 2 to the power of 3 multiplied by 3.

Did he get this right? Well done.

Now, for the next one, 36, I can start it as 36, and then think what are the two numbers that I know that multiply to give me 36.

I can write 6 multiplied by 6, and then I can go to the 6 and say, "Which two numbers multiply to give me 6?", and that is 2 multiplied by 3.

I circled both of them because they are both prime numbers, I do the same for the other 6.

Now, I can write down 36 as 2 multiplied by 2 multiplied by 3 multiplied by 3, or 36 is equal to 2 squared multiplied by 3 squared.

Now remember, your method may be slightly different to mine.

You may have started with different numbers.

I started with 6 multiplied by 6, you may have started with different numbers and that's okay, we should both end up with the same answer.

Wonder if you had this correct.

So the second part of the try this ask us to identify common factors.

I'm using a Venn diagram here to help me identify those common factors.

Please have a look at this diagram.

Have a little think about what does it mean.

Where did these numbers come from? Why did I circle some of the numbers? Just want you to have little think.

Okay.

Now, let's have a discussion about this.

Okay.

So I have here a Venn diagram.

I have one set called prime factors of 24, I have another set called prime factors of 36.

I know that 24 is equal to 2 multiplied by 2 multiplied by 3.

And I know that 36 is equal to 2 multiplied by 2 multiplied by 3 multiplied by 3.

Now, if I look at the prime factors of 24 and 36, I can see that we have some common ones.

For example, I have 2 that appears in both, I have another 2 that appears in both, I have a 3 that appears in both.

And these numbers are in the middle between the two sets, they are in the intersection area, the area where the two sets intersect each other.

And then I have a 3 left in the set of prime factors of 36, and I have 2 left in the prime factors of 24, and I have these separately in each set.

Now, this student here says, "4 is a common factor." It is a common factor.

Because if you look at the intersection section or part of the Venn diagram, you can see that we have a 2 and another 2.

2 multiplied by 2 is 4, so 4 is a common factor.

Now, is 4 the only common factor? Let's have a look.

I have 2 multiplied by 2 multiplied by 3 in the intersection part, which tells me that 12 is also a common factor.

I have 2 multiplied by 3 as a common factor, which is 6.

I obviously have 4, we discussed it.

I have 3 as a common factor between them.

I have 2 on its own.

And I have 1 as a common factor.

In fact, 1 is always a factor, but it's not visible in the Venn diagram.

So 1, 2, 3, 4, 6 and 12 are all common factors.

I can identify these from that intersection part of the Venn diagram, by taking these numbers and multiplying them.

Use a Venn diagram to find the HCF of 52 and 48.

We'll always start by finding the prime factors of the numbers.

So 52 is equal to 2 multiplied by 26, 26 is equal to 2 multiplied by 13.

So now, I can express 52 as 2 multiplied by 2 multiplied by 13.

I need to do the same for 48.

So 48 is equal to 2 multiplied by 24, the 24 can be written as 2 multiplied by 12, the 12 can be written as 3 multiplied by 4, and the 4 can be written as 2 multiplied by 2.

And this tells me that 48 is equal to 2 multiplied by 2 multiplied by 3 multiplied by 2 multiplied by 2.

Now, always remember that your factor tree could be slightly different, but we should end up with the same answers all the time.

Now, I can start to draw a Venn diagram.

I start with the box first, the square, just to show the whole set.

And inside it, I have two sets.

The first set is the prime factors of 52.

The second set is the prime factors of 48.

Now, I need to know where the numbers are going to go.

So I need to look at the prime factors of 52 and 48 and see if I have anything common.

I have the 2 in common.

Because I have 2 in common, I put it in the intersection section, so it's the section in the middle.

I only write it once.

Okay? If it's common, I only write it one time, and never have to write it twice.

Now, is there anything else common between them? Well, there is.

There's another 2 here.

So the 2 is also common one more time so I can write it again.

So notice here, I don't have four, lots of 2 in the intersection section, I only write it once if it's common.

Now, do I have anything else common? No.

The next thing I have is 13, and that is a prime factor of 52, only 52, so it goes here.

Now, I have 3 and I have a 2, and I have another 2, that's all prime factors of 48, so they go in the section for 48 for me.

And now, I have done my Venn diagram.

Now, the numbers that are in the intersection part, they show me the factors, the common factors, the common prime factors, but I want the HCF.

So to find the HCF, I need to multiply all of these numbers, it's the product of that common prime factors.

So this part here, the HCF tells me, the numbers that are in this part tell me the HCF.

So we have 2, so we multiply 2 by 2 and that gives us 4.

If you have three numbers, you have to multiply the three numbers.

If you have four numbers, it's the product of the four numbers and so on.

If you have one number, it's just that one number there.

So that HCF here is 4.

It is time now for you to have a go at the independent task.

For the first question, you need to write the numbers that I have given you as products of the prime factors.

Once you've done that, you use the answers from question one to answer question two, and complete the Venn diagrams. Once you've completed the Venn diagrams, I want you to find the HCF from each Venn diagram.

If feeling confident about this, please pause the video now and have a go at it.

If not, I would be giving you a hint in three, in two, in one.

Okay, and my hint is to start like this.

130 is equal to, think of two numbers.

The easiest one that I can think of is 10 multiplied by 13.

The product of 10 and 13 is 130.

13 is a prime number, I've circled it.

Now 10, I need to think of two numbers.

I need to do the same with the other ones, 104, it's equal to 2 multiplied by 52.

Then from 52, 2 multiplied by what number? Remember, when you have a prime factor, you circle it.

With this hint, you should be able to make a start.

The independent task should take you about 15 minutes to complete.

So please pause the video and complete the task, resume once you're finished.

Welcome back.

How did he go on with the independent task? Okay.

So I have my answers here.

I would like you to mark and correct your work.

130 is equal to 2 multiplied by 5 multiplied by 13.

104 is equal to 2 multiplied by 2 multiplied 2 multiplied by 13.

You may have written it as 2 to the power of 3 multiplied by 13, and that's also correct.

56 is equal to 2 multiplied by 2 multiplied by 2 multiplied by 7.

And the last one, 308 is equal to 2 multiplied by 2 multiplied by 7 multiplied by 11.

Did you get all of these correct? Really good job.

Now, with the Venn diagrams, I'm showing you here the answers to three of them.

Please mark and correct your work.

If you need to pause the video, please do so.

For the first one, the HCF of 130 and 104 is 26.

For the second one, the HCF 130 and 308 is 2.

And for the last one, the HCF of 56 and 308, was 28.

Did you get all of these correct? Really good.

Now, the third one, we're going to do together.

So I'm going to go through the solution for it.

To start with, I have 56 is equal to 2 multiplied by 2 multiplied by 7, and 104 is equal to 2 multiplied by 2 multiplied by 2 multiplied by 13.

I had my two sets, prime factors of 56 and prime factors of 104.

I have 2 common separate here.

I have another 2 common separate here.

I have the third 2 that is common, so I'm going to put it here in the intersection part.

I have a 7 left from the 56, and I have a 13 left from the 104.

And now, I've completed my Venn diagram, I need to write down the HCF.

And it's 2 multiplied by 2 multiplied by 2, which is equal to 8.

It's 2 times 2 which is 4, multiply it by 2, and that's 8.

Please make sure that you're really careful with this.

I have seen so many students making really serious errors here and writing 6.

So remember that it's the product of all the numbers.

Did you get this right? Really good.

Let's move on to question number three.

So this question number three, you had to place the prime numbers that you have been given in to the Venn diagram to find possible answers.

So there's more than one possible answer.

One of the answers could have been this one where you have 2 as a common factor, and it's actually your highest common factor because you don't have anything else in that, in that part.

Now, if I want to know what number this is, I need all of the numbers that are there, I need the product of all of these.

So 2 multiplied by 2 multiplied by 3 that gives me 12.

And then next set is all of that, there the product of all of those numbers.

So 2 multiplied by 5 is 10, multiplied by 7 is 70.

So I have now presented the information in the Venn diagram.

The two possible numbers are 12 and 70 with a HCF of 2.

But I could have also used the numbers to create completely different numbers, like the one here shown in blue.

I could have the 2 and the 2, so the two 2s, right in the middle, in the intersection part.

I could have put 3 here, 5 and 7 there, that would give me, in this case, 12 and 140, and tells me that the HCF of 12 and 140 is 4.

Now, I could've multiplied 2 and the 7.

I could'VE come up with so many different answers.

The most important thing is knowing that what we cannot do, we cannot have 2 here and 2 here.

Because if we have 2 in this part, as part of this set, and 2 as part of that set, then it's common.

In that case, it has to be in the middle.

So the only thing is you cannot have 2 in each set and no 2s in the middle.

You can have one 2 in the middle and one 2 in either sets.

I wonder what answers you came up with.

With our explore to ask, I would like you to select pairs of numbers, and find their highest common factor.

So you've got here six numbers, choose any two, find the highest common factor.

Then find another pair, so choose another two numbers, and find their highest common factor.

Then another pair and another pair.

How many pairs can you make? And what do you notice about the highest common factor of all of those numbers? If you really want to challenge yourself, create a similar set of numbers.

The explore task should take you between 15 to 20 minutes to complete, so please pause the video and complete it.

Resume once you're finished.

Welcome back.

How did you go on with the explore task? How many pairs did you make? What was the highest common factor of your numbers? Really interesting.

So you could have made 15 pairs in total.

If you paired the numbers, you could have made a maximum of 15 pairs.

This is what I noticed.

The highest common factor of 24 and 72 was 24.

The highest common factor 24 and 48 is also 24.

The highest common factor of 24 and 120 is 24.

The highest common factor of 24 and 168 is 24.

Some of you may think here, just because we have the 24 in one of our numbers, that's why the highest common factor is 24.

So we'll try it with different numbers.

The highest common factor of 72 and 264 is 24 as well.

So what we have noticed here is the highest common factor of all 15 pairs, no matter which two numbers you chose here, you should have ended up with a highest common factor of 24.

Did you do this for all 15 pairs? Did you create a similar set of numbers that have one or the same highest common factor? I wonder what you've done.

This brings us to the end of today's lesson.

A huge well done on all the effort that you've made.

Please remember to complete the exit quiz to show what you know.

Enjoy the rest of your day, and I'll see you in another lesson.

Bye.