Sherpa This AQA GCSE Maths video covers factors, multiples and prime factorisation. It looks at how to find the factors of a number, what is a multiple of a number and how to find the prime factors of a number.

The video also looks at square numbers and cube numbers and how to use the prime factor tree to help find the LCM and HCF of two or more numbers.

A multiple of a number is any integer multiplied by the number.

A factor is an integer (whole number) that will divide exactly into another number.

Factor pairs are two numbers which multiply together to make a particular number.

We can split up factors and multiples into five smaller buy size chunks. So we've got factor pairs, square and cube numbers, prime factor trees, lowest grow, multiple and highest common factor. So if you look at factor trees first, sorry, factor pairs first. Let's do an example. So let's start off with say, 24.

That's got quality factors. And all we're going to do is we're going to list out the pairs of numbers that times together to give us 24. So I know one times 24 gives me 24. I can do two times twelve or I can do three times eight where I can do four times six and I can keep going, right? But what would see happening is I basically end up mirroring what I've just done, right?

Say I keep increasing these numbers, but we actually end up with like a mirror line and anything below that mirror line is just a repetition of what we've done previously. So these are really my four pairs, my factor pairs, just numbers that time together to get the original number. In this case, 24. Square keep numbers. So square numbers are formed when we take a number and we times it by itself.

So for example, four is going to be a square number. Or I could do five times five and I could get 25. And again 25 is there for a square number because a number of times itself gives us a square number. Cube numbers are very similar. In this case, we just times the number by itself three times.

So two times two times T would give us eight. So eight be a cube number. And continuing with example, I do five times five times five or five times five is 25 times five would give me 125. So these would be my coup numbers and these would be my square numbers.

Just examples of this. I'm going to take prime facts trees over here. So prime facts trees are where we basically break down a number into smaller and smaller numbers until we can't break it down anymore. So for example, let's continue on 24 as an example. So I split 24 up into two numbers and normally we just pick two and something else.

In this case it's going to be two times twelve. No, but two times twelve from my practical pair gives me 24. Well, I can't break up two into anything smaller, right? If I tried to do that, I'd get 21212 one continuing on forever. So as soon as we get to a prime number that can't be still up anymore, that's the end of that branch where you keep getting with twelve.

I know two times six gives me twelve and I can split six up even further. I know two times three gives me six and I can work my way back up this tree, right? I know that two times three gives me six. So I know that if I do two times three, I get six. And I know that six times two gives me twelve.

So this whole thing is twelve. And I know that twelve times T must give me 24. So this whole thing gives me 24. And so I can times together my prime factors of a number and I can get back to my original number. So I can always represent your original number as the product of his prime factors.

Moving on, well, I'll go into more depth in a second with low score factor to stick with a nice example. For now, we can use the prime factor trees to help us sell these, which we'll do in a second, but if I just pick a nicer example to prove the concept, let's pick 18 and twelve when I find the lowest con multiple between 18 and twelve. So one way of doing this to kind of brute force it is to just list out the multiples of 18 and twelve. And I could keep doing this, I end up with 54 I think would be the next multiple and so on with twelve, and I get 1224, 36, 48, and again, so on. And what we can see is we've got this common factor between both 36 would be the common factor between 18 and twelve.

Moving on to the highest common factor. Again, if we look at this very briefly, for now, I'll stick with the same example of 18 and twelve, and I'm trying to find the biggest number that goes into both 18 and twelve. And so I could go back to my factor pairs topic and I could look at the factor pairs of 18 and twelve. Well, I know one times 18 is 18, two times nine is 18, three times six is 1824, 25. Then I get to six and I repeat what I've just done.

Right, that's the mirror line there. And twelve is very similar, one times twelve, or I can do two times six, or I can do three times four. And again, once I get to four times three, I'm just repeating myself. Well, I know that if I look at these numbers, I'm trying to find the biggest number that they both share in these pairs. I can spot here, I've got a six here and a six here, I've got a nine in 18, there's no nine in twelve.

I've got twelve and twelve, but there's no twelve and 18. So six is the biggest number for the bracher. But again, we'll see in a second how we can actually use the prime factor trees to tackle more complex versions of this topic. All right, let's dive into some exam style questions. So, got a list of numbers and we're going to write down a model of six.

So anything in the six times table, well, that's just going to be twelve if we know our six times tables factor of 15. So anything that goes into 15, a whole number of times. Well, I know five goes into 15 three times, so that's going to be a factor of 15 square number. So a number that's formed when we multiply a smaller number by itself. Well, I know that three times three gives me nine and nine is in that list, so nine is going to be my square number.

Prime Facts trees. So we'll do another quick example. This time we'll look at 72. So we're going to prep ourselves for the questions to come. So I can split this up into say, two and 36, which in turn goes to two and 18, which in turn goes to two and nine, which in turn goes to three and three.

So I know that I can do three times three to get nine. I can then times that by T and that gives me 18. I can times this whole thing by T and that gives me 36 and I can times that whole thing, this whole result by T and that gets me back to 72 as well. So this whole thing would then give me 70, so that gives me nine. The next up gets me to 18, the next up gets me to 36 and the entire thing gives me 72.

And we can be smart by this. We've got three times three and two times two times two. So we could also write this as well. I know three times three can be three squared and two times two times three times two can be written as two cubes. Let's go into the hard stuff.

This is the highest point factor, a few numbers. So again, I'm going to do my factory trees. You'll see 1 second for both these two numbers or 60, I can set that into 32, then I can go 15 and two, then I can do five and three and I can't split these up any further. So when I get to prime, that's the end of the branch. Likewise with one month four, if I split that into two, that's going to give me 57 and I can then split that into multiple of three.

So I can split into three and 19 and 19 is prime. So I know I've reached the end of all my branches. And what we're going to do now is we're going to look at the common factors between these two numbers. So I've got a two here, I've got a two here and a two here. So I know two is a common factor of both these two numbers.

Well, before anything else, I can actually write down 60, the product of those prime factors, just like we did above. I know 60 is just two times two, times five, times three, right, but times all those prime factors together, it doesn't matter which way I'm going to go. I'm actually going to for the purposes of this, to show you what's going on. But again, it doesn't matter. And I know that 14 is two times three times 19, so 14 is just two times three times 19.

Well, what I can see is I've got two times three appearing in both of my numbers, right, which is six. I know six therefore goes into both my numbers. So whenever I've got a common pair of numbers, I know that I can times them together and they'll then give me I can times all my pairs together and they'll then give me the highest common factor. So we don't have any other pairs, right? 19 doesn't pair with anything here, so we've only got that two pairing without two and that three pairing without three.

And so I know that I've got a two and a three occurring in both of these products of prime factors. Right, final question, finding the lowest point multiple between 17 and 50. So again, we're going to start off doing the prime factor trees again. You'll see why in a second this is going to give me seven and ten, and ten is going to give me five. And because seven is prime, that's the end of that branch.

Likewise, 56, I can get seven and eight, I can get two and four and I can get t and T. So those are my prime factors, get 56. Well, let's go to trying to find the lowest coin multiple, right? We can think of this as like acorn multiple.

Very easy way of doing it. It's just 70 times 56, right? Because we know that 70 is going to go into it 56 times and 56 is going to go into it 70 times. Well, we also know that 70 can be written as seven times t times t times t. So I'm multiplying this, which is seven t, by 56, which is seven times t times t times t, and that gives me 56.

What we can see is we're getting this multiple, but it's likely bigger than we need it, right? There's probably a smaller number that both 17 and 56 will also go into. And what we're going to do is we're going to try and spot where that redundancy is, where we're introducing more numbers than we need, right? And what we can see is we've got seven in here and we've got a seven in here that we can actually share one seven between these two numbers. And likewise, you've got a two here and a two here, but these two numbers can actually share just one t between them.

We don't need two t's. So what I'm going to do is I'm going to get rid of one of each pair and if I rewrite this, I'm going to rewrite this over here, I'm going to have five times seven times t, times t, times t. So I get five times seven times t, times t, times t. And what we can see is we've still got 70 appearing, right? We've still got 70 appearing here because I know that five times seven times t must give me 70.

And I've also got 56 appearing, right? I know there's 756. So by getting rid of one of each pair, I've kind of gotten rid of all the redundancy, and I've reduced this just general coin multiple number down to the smallest coin multiple. This is my LCM. And so we always get rid of one of the two pairs.

If you think of that on the prime factor trees, I can identify my pairs. I've got a two here and a two here. I get rid of one of those. I've got, say, seven here and a seven here. I get rid of one of those, and all I'm left with is five times seven, which is five times seven times t times, times T.

And so that's how we could do it directly on our factory. We just cross out one of each pairs. That is the end of factors and multiples.

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