Lesson video

Hi, 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.

Today's lesson's title is Lowest Common Multiple and it features in the unit properties of number.

So by the end of this lesson, we will be able to use prime factorization of two or more positive integers to efficiently find their lowest common multiple.

So for example, what is the lowest common multiple of 18 and 52? Key words that we'll be using in today's lesson.

The lowest common multiple is the lowest number that is a multiple of two or more numbers.

LCM is an abbreviation for lowest common multiple.

So from here on we can refer to it as LCM and we will all know what that means.

Today's lesson, we are going to split into three learning cycles.

The first one is going to be looking at common multiples using prime factorization.

Then we'll move on to looking at the lowest common multiple using prime factorization.

And then finally, the lowest common multiple of more than two numbers.

So let's get us going on common multiples using prime factorization.

How do we find the first five multiples of a number? Well, the first multiple is one multiplied by that number.

So let's have a think, multiples of three.

So it's one multiplied by three, which is three.

The second multiple is two multiplied by the number.

So the second multiple of three is two multiplied by three, which is six.

The third multiple is three multiplied by the number which is three multiplied by three, which is nine.

The fourth multiple is four multiplied by the number that gives us four multiplied by three, which is 12.

And finally the fifth multiple is five multiplied by the number, five multiplied by three is 15.

Do you think we could find the first five multiples of two squared multiplied by three? We can in the same way.

The first multiple is going to be one multiplied by two squared multiplied by three.

And we know that multiplying by one does not change the number we've started with.

So it's two squared multiplied by three.

The second multiple then would be two multiplied by two squared multiplied by three.

So two multiplied by two squared multiplied by three.

That simplifies to two cubed multiplied by three 'cause if we look, two multiplied by two squared is actually the same as two cubed.

The third multiple would be three multiplied by two squared multiplied by three, which is two squared multiplied by three squared.

So notice again we've simplified the left hand side, three multiplied by three to three squared.

Now let's look at the fourth multiple.

So the fourth multiple is four multiplied by two squared multiplied by three, which is two squared multiplied by two squared multiplied by three because remember four is two squared, which is two to the power of four multiplied by three because two squared multiplied by two squared is two to the power of four, be a repeated multiplication of two four times.

And finally we'll look at the fifth multiple.

So the fifth multiple would be five multiplied by two squared multiplied by three, which is two squared multiplied by three, multiplied by five.

Notice I've just changed them so that they are in numerical order.

Let's now think about the multiples of two multiplied by three multiplied by five.

So the first one would be one multiplied by two, multiplied by three, multiplied by five, which is two multiplied by three multiplied by five.

And then the second one would be two multiplied by two, multiplied by three, multiplied by five, which is two squared multiplied by three, multiplied by five because again, two multiplied by two, we can simplify to two squared.

The third multiple is two multiplied by three squared multiplied by five.

Again, we've simplified the two, the multiplication of three twice to three squared.

The next one would be the fourth one, which is two squared multiplied by two, multiplied by three, multiplied by five.

So remember here we've replaced the four with two squared, and then we can simplify that to two cubed multiplied by three, multiplied by five because two squared multiplied by two is the same as two cubed and then five multiplied by two, multiplied by three, multiplied by five is two multiplied by three multiplied by five squared.

And again, there we've simplified the left hand side into the right hand side.

So here are our multiples of those numbers.

We notice here that this is a common multiple.

So two squared multiplied by three, multiplied by five appears in both lists.

So therefore, it is a common multiple.

The multiples as well, don't forget that the multiples of that will always become a multiples of the two original numbers as well.

So the multiples of two squared multiplied by three, multiplied by five will be always be common multiples of two squared multiplied by three, multiplied by five and two multiplied by three, multiplied by five.

So for example, two squared multiplied by three, multiplied by five squared.

We could find any number of them that we wanted to.

So here are some examples and non-examples of multiples of two squared multiplied by three squared, two squared multiplied by three squared multiplied by two.

We can see that that is actually the second multiple of two squared multiplied by three squared.

Two squared multiplied by three squared multiplied by five.

Well there's our original product of prime and we can see that this must then be the fifth multiple.

Two squared multiplied by three cubed multiplied by five squared.

So we were finding multiples of two squared multiplied by three squared.

So let's start with that.

And then what do we need to introduce to that product to make it equivalent to the left hand side, we would need a three and a five squared.

And then we could see that if we do three multiplied by five squared, that this is the 75th multiple.

In order to be a multiple of a number, then that number must be a factor.

So two multiplied by three squared, two squared multiplied by three squared is not a factor.

These are all non-examples.

There's another one and another one.

We're now going to look at using our prime factors to find common factors.

So we've got 30 is two multiplied by three, multiplied by five, and 18 is two multiplied by three, multiplied by three.

What is the same and what is different about these two products? Pause the video and write down what you think is the same and what you think is different about the two products.

So some examples of things that you might have written down.

Both of them have factors of two and three.

Both of them are the product of three integers.

The third integer is different in each of them.

So they've both got the common two multiply by three.

But we can see that that third factor is actually different in both of them.

What is the same and what is different about these three products? So I want to again, pause the video, write down for me what's the same and what is different about these three products.

So some examples of things that you might have written down.

They're all products.

Well, we can see that, they've all got a multiplication in them.

The top two are products of three integers and the bottom one is only two.

The top two only use prime numbers, but the bottom one uses a composite number and a prime number.

So those are some examples of things that you might have thought about and written down.

How can we rewrite the third product to be more similar to the original two? Well, 114 is six multiplied by 19, but six actually we can break that down into its prime factors, which are two and three.

So we could write this as two multiplied by three, multiplied by 19.

Now each of those is written as a product of just prime factors.

These are all multiples of six.

Write them as products of two multiplied by three.

You can have a go at these before I go through them if you want to, you could pause the video and have a think about it.

If you're not too sure though, you could just stick with me.

And we're gonna go through the answers to these in a moment.

Starting with 12, then.

Six multiplied by two.

That's our factor pair, that's where I'm gonna start.

But we know that six can be written as two multiplied by three.

So remembering as well that we can write two multiplied by two in its exponent form.

So two multiplied by two is two squared.

78, now let's think about 78.

So 78 is six multiplied by 13.

We know that the six could be rewritten as two multiplied by three.

And then the final one, again, 102 is six multiplied by 17.

But we know that six can be written as two multiplied by three.

We've now written all of those as a product of it, prime factors.

Now let's have a check for understanding.

I'm pretty certain you'll be ready and good to go for this one.

So it's a true or false, two cubed multiplied by three squared is a multiple of two squared multiplied by three squared.

So as always, I'd like you please to decide whether you think the answer is true or false.

But don't forget the most important part of this is your justification of your answer.

Why have you chosen true or false? Pause the video and then pop back when you're ready.

Brilliant, well done.

So, this was true.

So the answer to this was true and my justification is A, two cubed multiplied by three squared is the same as two squared multiplied by three squared multiplied by two.

So it is the second multiple of two cubed multiplied by three squared.

Well done if you've got that right.

Brilliant, so we are now ready to have a go at some independent work.

So task A.

So I'd like you to write the first eight multiples of the following as products.

And then for the last one, you are gonna work out the first common multiple of 15 and 12.

Pause the video and come back when you're ready.

Okay, let's check these answers then.

So the first eight multiples of the following.

So the first one remember is just the product and then two multiplied by the product, three multiplied by the product, et cetera, remembering to simplify any prime factors where they appear more than once and putting them in their exponent form.

I'm not gonna read through all of those.

You can pause the video once I've put up the answers to A, B, and C and then come back when you've marked them.

So here are the answers to B and the answers to C.

So pause the video, check off those answers, and then you can come back.

Super work.

So final one, D, the common multiple is two squared multiplied by three, multiplied by five, which is 60.

We are now ready to move on to our second learning cycle, lowest common multiple using prime factorization.

So up until now, we've just been considering common multiples.

We now want to think about what's the lowest common multiple.

What would be the lowest common multiple of 24 and 40? So I've given you there both of those integers written as a product of its prime factors.

We need to remember that a factor multiplied by a factor gives us a product.

So therefore a factor multiplied by a factor gives us a multiple.

For an integer to be a common multiple of both numbers, it must be a product of two common factors.

So two cubed multiplied by three and two cubed multiplied by five.

Those are our two products of prime factors for 24 and 40.

So we can see two cubed is common to both of them.

Now I need to create my second factor so that it is common to both of them.

So if I look at here, I've got a three and in this one I've got a five.

How am I gonna make those two things the same? Well, the top one I need to introduce factor of five, and for the bottom one I'm going to need to introduce that factor of three.

So the lowest common multiple of 24 and 40 is the product of those two things, two cubed multiplied by three, multiplied by five, which is 120.

We could also do this using a Venn diagram.

And personally I normally prefer to use a Venn diagram.

I like to see the visualisation of what's going on.

So let's put these into our Venn diagram.

So two cubed is common to both.

So it's going in the intersection, that middle part of the diagram.

What was missing from the first one was just the factor of three and then the second one was five.

What do you notice about those prime factors? Look at how we calculated 120 and look at the Venn diagram.

Do you notice anything? Hopefully you noticed that the lowest common multiple, the LCM is the product of all of the prime factors in that Venn diagram.

That's why I quite like the Venn diagram 'cause it does make it easy to see what numbers we are finding the product of.

Let's put that into practise.

So we're going to find the lowest common multiple, the LCM of 360 and 2,100.

So here they are as a product, their prime factors.

I'm just gonna write one below the other and we're going to put these into our Venn diagram 'cause like I said, I prefer to do it that way.

Remember you could go back and do it without the Venn diagram.

That's absolutely fine, that's personal choice.

So starting with two, we can see here we've got two cubed and two squared.

What's common to both is two squared.

So that's gonna go in the intersection of the diagram.

The top number though, 360, needs an additional factor of two.

So we're gonna put that into the left side of the Venn diagram.

Let's move on to our prime factor of three.

So there's our threes.

So what is common to both of them is just this time of three.

So let's put that into our product and then put that into our Venn diagram.

The top number, 360, needs to be three squared.

So we are missing a factor of three and then we're gonna pop that again into the diagram.

Now we're gonna move on to five.

So our common factor here is five.

So let's put it into our two numbers and sorry, our two products, and then into our Venn diagram.

2,100 actually is five squared.

So we need to introduce that second factor of five and put that into the Venn diagram.

And then finally we look at the seven and we can see that seven is only in 2,100.

So that's going to go in the right hand side of the Venn diagram.

So we can see here what is common to both.

These are common to both.

I need to make this factor the same as this factor.

So what do I multiply two and three by to make it match five and seven? So that's just the five and the seven.

And then the other side is going to be the two multiplied by the three.

So the LCM of 360 and 2,100 is two cubed.

So notice here I've written it as two cubed because I know that two squared multiplied by two is two cubed, then three squared from the Venn diagram multiplied by five squared multiplied by seven.

The lowest common multiple of those two numbers is 12,600.

We're now gonna have a go at one more question together and then I know that you'll be ready to have a go at one of these independently.

You might even feel ready to have a go at it independently now and choose to pause the video and have a go at it.

But if you are not too confident, you might want to just have this as a final little example to go through before you try one independently.

So we are going to find the lowest common multiple of 300 and 504.

So I've given them to you again as a product of their prime factors and we need to put them into the Venn diagram.

So using our system, starting with the lowest prime number, which is two, what's common to both two squared.

But 504 is actually two cubed, so we need to put the other two in.

Notice this time I'm only using the Venn diagram, that's 'cause I find it most useful.

Then three, we've got a three and a three squared.

So common here is a three, but 504 is actually three squared.

So we need to introduce that extra factor of three into just the right hand side of the diagram.

Then we'll move on to our next prime number, which is five, five squared.

There are no fives in 504, so that's just gonna go in the left hand side of the diagram.

And then seven, seven is only in 504.

So that is going to go in the right hand side of the diagram.

So remember to find the lowest common multiple, we need to find the product of all of the factors that we've put into our Venn diagram.

So the LCM 300 and 504 is the product of all of those.

So two cubed, again, I've just written it in its simplest exponent form, two squared multiplied by two, multiplied by three squared, multiplied by five squared, multiplied by seven.

And of course you can use a calculator to do that.

I don't want you sit there scribbling away doing that without a calculator.

Use your calculators now, which is 12,600.

Now you'll be ready to have a go at one of these independently.

But don't worry if you are not, remember, you could always go back and re-watch the examples.

So the question that I would like you to have a go at is find the lowest common multiple, the LCM of 648 and 540.

Here are the numbers written as a product to their prime factors.

This is what your Venn diagram should look like.

So have a go at this question.

Remember you can use a calculator for that final step, finding that final product.

So pause the video and come back and check in with me to see if you've got your answer right.

Great work, let's have a look.

So this is what your Venn diagram should look like.

So the lowest common multiple of 648 and 540 is the product of two cubed multiplied by three to the power of four multiplied by five, which is 3,240.

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

We can now move on to task B.

So task B, you are going to use the Venn diagrams and the product prime factors to find the LCM of each of those pairs of numbers.

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

Okay, question number two.

So this is a lot more challenging this question, but I know you are up for a challenge and that you are going to give this a real good shot.

The LCM of 420 and another number is represented in the diagram is 1,260.

So I'd like you to have a go and work out what is the missing prime factor and what was the other number? And again here you can use your calculator.

Pause the video, give this a real good go, and then come back when you are ready.

Wow, that was quick.

Let's check our answers then.

Sure you got those right, well done.

If you didn't just have a look and see.

Can you see what your error was? Was it that you'd put the numbers into the Venn diagram incorrectly or maybe you just mistyped something when you were finding the product.

So it's always worth checking that what you've written into your calculator is what you were expecting to find the product of.

And then the second one, like I said, absolutely superb if you've got this right 'cause it was challenging.

The correct answer to A was three.

There was a three missing.

So we needed to work out what two squared multiplied by three, multiplied by five, multiplied by seven was, and then we needed to work out what that missing prime factor was to give us 1,260.

The other number would be the product of the numbers in the left hand set.

Now we can move on to our final learning cycle.

So we are now going to be looking at what happens if we want to find the lowest common multiple of more than two numbers? So you've done this brilliantly up until now.

You can find the lowest common multiple of two numbers.

We'll now move on to looking at three.

And actually it's just as easy as doing two, I think.

We're gonna find the lowest common multiple of 84,900 and 225.

So here's our Venn diagram and here they are as a product of their prime factors.

So let's go through them.

So we've got two is our lowest prime number.

So two squared, that's common to 84 and 900.

So it's going to go here.

Now let's look at three.

What's common to all of them is three.

So that's gonna go in the centre and then we are missing prime factor of three in 900 and three in 225.

So that needs to go in that spot.

Now let's look at five.

So five squared, we can see five squared is common to 900 and 225.

So it needs to go in the same place as the three.

And then finally the seven, well the seven is only a factor of 84.

So it's just going to go in a factor of 84.

The lowest common multiple therefore of those three numbers is the product of all of the numbers in the Venn diagram.

So again here, just like I did before, I've written three multiplied by three 'cause it appears twice as three squared.

So the LCM of those three numbers is 6,300.

Now if you're feeling confident after that example, you might decide you are up for a challenge and you are going to pause the video and have a go at this one first.

But if you are not so confident, you might want to go through this one with me first before you try it one independently.

We're going to find the lowest common multiple of 198, 270, and 1,350.

Here's our Venn diagram and here are our product of prime factors.

So we're gonna start with two.

So two we can see is common to all three.

So we're going to put that in the centre of our Venn diagram.

Now let's move on to three.

So we've got three squared, three cubed, and three cubed.

So common to all of them is three squared, but we are missing a factor of three from 270 and 1,350.

So that needs to go in the intersection of just those two.

Let's now move on to five.

So we've got a five and a five squared.

So the five is common to 270 and 1,350.

And then we need that extra factor of five for 1,350, and then finally 11.

And we can see that that is only a factor of 198.

So that is gonna go just in a factor of 198.

So now to find the LCM, the lowest column multiple, we find the product of all of the factors.

And here I've just written each of the prime factors in its simplest exponent form.

So two multiplied by three cubed, multiplied by five squared, multiplied by 11.

And again, I'm not expecting you to work that out without a calculator.

So the answer to that is 14,850.

Well done if you challenge yourself to have a go at that before I went through it, that's absolutely superb.

And even better if you managed to get the answer correct, but don't worry if you didn't.

Now we're ready to do a check for understanding.

Here we have Alex, do you agree with Alex? Alex has actually found the HCFs, the highest common factor and not the LCM, and that's a common mistake.

He should have found the product of all of the factors in the Venn diagram.

Now you are ready to have a go at a task yourself.

Now, just as question two in task B was a challenging question, this is also challenging, but I have every confidence that you'll be fine and you'll be able to complete this.

So you'll need to fill in the blanks.

So pause the video and then when you are done, come back and we'll check and we'll see how you've got on.

Let's see how you got on.

So in 180, we were missing a three.

Then the missing integer was 140 on the second line, and then the missing integer and the bottom line was 660.

And then in the Venn diagram, just in 180, we were missing three.

In the intersection of all three, we were missing two squared.

And then just in the 660, we were missing 11.

Absolutely amazing if you got that right.

It was a particularly challenging question, but I know that you are up for a challenge.

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

So we looked at first common multiples can be found using prime factorization.

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

But like I said during the lesson, I'm an advocate of Venn diagrams I find really useful.

And there is an example there so that you can refer back to that.

You've done amazingly well persevering with what was some quite challenging things today.

So well done.