# Lesson video

In progress...

Hello, I'm Mr Coward and welcome to today's lesson on lowest common multiple.

For today's lesson, all you'll need is a pen and paper or something to write on and with.

If you can, please take a moment to clear away any distractions, including turning off any notifications.

And if you can, try and find a quiet space to work where you won't be disturbed.

Okay, when you're ready, let's begin.

Okay, so time for the try this task.

You've been told that this rod that covers four squares has a length of 60.

Can you find the value of each of the remaining rods? What if this wasn't 60 or what if you're told a different number, could you work out the value of the rods then? So have a go for a different number and try.

So pause the video and I have a go in three, two, one.

Okay, welcome back.

Now you can see that this one is four squares.

So four squares that add to 60 which means we can do 60 divided by four to find one square.

15 plus 15, 30 plus 15 plus 15, 60.

Ah, so 15 plus 15, two squares that would be equal to 30.

So what does that mean for three squares? Whereas 15 more than that, or 15 less than that.

45 and then one more square.

So one more 75.

So really well done if you've got those correct.

And really well done if you tried for a different rod.

So if you said, hmm, what if this one was, well, this one here, what if it was 12? What would my other rods be? You hopefully spent some time exploring the proportional relationship between them.

Okay, help Yasmin to find the common multiples of 45 and 60.

So I want more than one common multiple.

And what is the lowest common multiple? So when will these two line up? When will they have multiples in common? So pause the video and have a go, pause in three, two, one.

Okay, so let's imagine this is like three.

So we've got another one here, so another 45.

That is that far, so there's another 60.

And then there, we've got another 45.

So they lineup then and let's try again.

We've got 45, 45 there, here we've got another 60, another 60, another 60, another 45 and another 45.

So they line up then, what number is that? 60 plus 60 plus 60, 180 plus another 60.

And another an another 360.

And you could check 45 plus 45, 90.

Plus another 90, 180.

So you can see that they line up at 180 and 360 and they would actually line up for every 180.

So at another 180 they'd line up again.

At another 180 they'd line up again.

Now the first time they line up, we call that the lowest common multiple because all these numbers, all these lengths so 45, 90, 135, 180.

These are our multiples of 45.

And here 60, 120, 180, 240, et cetera.

These are our multiples of 60.

Now when they first multiples, when they first meet, we call that the lowest common multiple.

So there is lots of multiples but when they first meet, the first multiple they have in common, okay.

Think of it, it's telling you the word, the lowest, the lowest number.

So which of these is lowest? Well, 180.

It must be a multiple.

These are the multiples of 45.

These are the multiples of 60 and it must be in common.

So it must be both a multiple of that and a multiple of that.

So find the lowest common multiple of the following sets of numbers 18 and 30.

Well, what are the multiples of 18? We'll have 18, 36 double it.

Add on another 18 to this.

So when I added on 18, I can add on 20, 56 and takeaway two, 54.

Well, when I double this one to get four times 18, 72.

Then I'm going to add on another 18 to get five times 18, 90, 108, 126.

And I'll stop for there.

And I'll see what my multiples of 30 are.

So we've got 30, 60, 90, 120, et cetera.

Can you see one that's in common? 90 and 90 is our first multiple in common.

So 90 is our lowest common multiple which we sometimes write like this.

It's just a bit lazy, really lowest common multiple.

So the LCM, and that's how we say it.

The LCM of 18 and 30 equals 90.

So that means the lowest common multiple.

And we say it like LCM.

So the LCM of 18 and 30 is 90.

Okay, another one.

So what are multiples of 11? Et cetera.

And what are our multiples of five? Do we have one in common yet? No, so we have to keep going until we have one in common.

So, we can keep going but I'm actually not want to keep going 'cause I can see that ends in a one, that ends in a two, that ends in a three, that ends in a four.

So they're not going to be in my five-times table are they? But this one, this ends in a five.

Is that in my five-times table? Well, what would I have? I'd have 50 and then 55.

And I would not have any of these numbers in my five-times table 'cause they don't end in a zero or a five.

So this one is my lowest common multiple.

So you can sometimes use short-cuts like this by knowing your multiplication facts to help you find the lowest common multiple.

So the LCM is 55 but just when you're doing this, just be careful that you don't miss any multiples in common.

So you need to be really sure that none of the ones below it, would meet sooner.

Pause the video and have a go, pause in three, two, one.

Okay, welcome back.

Hopefully you found the multiples of 20, et cetera, the multiples of 25.

Oh and we could have stopped there.

Why could we have stopped there? Because we have the first, the lowest common multiple.

Okay, awesome so that's our lowest common multiple.

Okay, I've got 10 there and then nine, 18, 27, 36.

And, oh, we've got a 36 in common there.

Okay, so our lowest common multiple of four and nine equals and some people actually writes like this and I think that's quite smart way of writing it.

So the lowest common multiple of four and nine equals 36.

So that is a nice way of writing the lowest common multiple like that.

So anyway, now you see this one and this one.

Can you see how they're similar and different to these two? Well, what if I tell you that this is the fifth multiple of 11 and this is the 11th multiple of five.

And this one is the ninth multiple of four and the fourth multiple of nine.

Whereas over here, that's the fifth multiple, and that's the fourth.

That's the fifth.

And that's the third.

What's different about these? Well, these are co-prime and that means they have no common factor.

So when two numbers have no common factor.

Okay, so that 11 and five, there's no number of the one that goes into 11 and five.

So because they have no number in common, the lowest common multiple is actually just 11 times five.

What about this one? Do four and nine have a factor in common? No, four and nine do not have a factor in common.

So four times nine is 36 and that's our lowest common multiple.

Whereas over here, these do have a factor in common.

What is the factory in common there? Five.

Okay, so our lowest common multiple because they have a factor in common is not going to be 20 times 25.

It's going to be affected by that five.

And there's actually a very special relationship there that I don't want to tell you too much about right now.

But there's a relationship between the highest common factor of these two numbers and the lowest common multiple of these two numbers and the product of these two numbers.

So maybe this is something you might want to explore on your own.

What is the relationship between 20 times 25? The highest common factor of these two numbers and the lowest common multiple of these two numbers? So that may be something you want to spend a little bit of time thinking about and try it for this one as well.

See if what you think happens for this one.

What does it happen for these? Does it happen to another set of numbers? What about if you think of this, having a highest common factor of one, does it now work for this? Okay, so you might want to have a little think about that.

So it's time for the independent task.

So I'd like you to pause your video to complete your task and resume once you're finished.

And if you want to have a little think about what problem I paused you then, please do so.

Okay, welcome back.

You may need to pause the video to mark your work.

Okay, so we're with three.

There was lots of different possible numbers that have these lowest common multiples.

So if you didn't get these, it's fine.

Okay, so now it's time for the explore task.

So select two of the numbers and find their lowest common multiple.

So choose any two of these and find their lowest common multiple and then repeat for every possible pair.

Now I've given you some squared paper in your explore task but if you haven't printed this off, that's fine.

You might just want to use your square paper or you might just want to use some normal paper and just be a bit more accurate with your square drawing.

And then once you've done this for all possible pairs, choose a different set of numbers and see if you notice anything.

So pause the video and have a go and resume once you've finished.

Okay, so here are my answers.

Now I have 30, 45, 60.

So 60 is the lowest common multiple of 15 and 60, 75, 90, 300 is the lowest common multiple of 75 and 60.

Now what I want you to think about and you would have saw this if you could tried different numbers, is that say for instance, you said that the little one was 10 which means that that was 20.

You would have found these exact same combinations, no matter what numbers you tried.

So as long as you labelled the rods correctly.

So if one of them, the white one was 10, the red one would be 20, the green one would be 30, et cetera.

So if you labelled them correctly, you would have always found this exact combination.

This exact picture but just with different numbers.

Now why? Well, this one is two times bigger than this one.

So they will always meet like that.

This one, the purple one is four times bigger than that one.

They will always meet like that.

This one is two thirds of the size of that one.

So two of them will always be the same as three of them.

They have this relationship, this proportional relationship and that affects what their lowest common multiple is.

And this is a really important idea.

And it's also to do with the highest common factor as well.

So this one has a highest common factor of 15.

Can you see how 15 impacts these? Well, that's two 15, three 15, four 15, six 15.

So that's six times 15.

This one, this one is four times 15.

Sorry, this one is still six times 15 but it's been split differently.

So we can say that this one is two times three times 15, three times 15 is 45.

Whereas this one it's still six times 15, but we're actually doing three times, two times 15 because each rod is 30.

So we've got the same multiplication here but it's just written in a different way using associativity and commutativity.

And now this has an impact on finding our lowest common multiple.

And you might want to have a little think about this and see if you can write 75 in terms of 15 and write 45 in terms of 15.

In fact, let's do another one.

Let's do the 75.

So 70, well that's five times 15, right? And then times three.

45 is three times 15 times five.

Can you see how they are the same? So can you see where the number of rods is coming from? Can you see the highest common factor of these two numbers? Well, it's both 15.

So can you see how what we have left five and a three? They combine five times, three times 15.

They combine to give us 225.