# Lesson video

In progress...

Hello, I'm Mr Langton, and today we're going to look at prime deductions, how we can use our knowledge of prime factorization to deduce the factors of numbers.

All you're going to need something to write with and something to write on.

Try and find a nice quiet space where you won't be disturbed and when you're ready we'll begin.

How many different bouquets of flowers can you make using the four flowers below? Pause the video and have a go.

When you're done unpause it and we can go through it together.

You can pause in three, two, one.

How did you get on? I've come up with 12 different ways that we could do it.

Now I've also included the fact that I could have an empty flower pot.

I could have one of each type.

I could have two of each type, including two yellows in one of them.

And also looking at having three in each one and all four in each one.

Anthony says that there are only three factors of this number, two, five, and seven.

And we can see those up there, can't we? It's two, five, and seven.

Yasmin thinks that there are lots more factors.

Who do you agree with? Well, let's start off by looking at all the different possible combinations that we can make.

So, to factor this number, certainly include two, five, and seven.

Well we've got five squared there, so 25 is also a factor.

We're got two lots of five, which is 10.

That makes it a factor.

We could have two lots of 25.

Two lots of 25 is 50, so 50 must be a factor.

We could have two lots of seven, which is 14.

We could have two lots of seven multiplied by five, which is 70.

And that's just the start.

There are loads of factors that we can make for this number.

And just out of interest, that number is going to be, five squared is 25, 50, the number that we're working with is 350.

Although there are three different prime factors there are loads and loads and loads of factors that we can make, so Yasmin is correct.

And let's look at how we can use this method to find all the factors of 40.

Let's make a little space over here.

So, 40 is made up of five times eight, which is made up of two times four, which is made up of two times two.

So I've got two cubed times five to make 40.

So what could my factors be? If I start off with two.

And I could have two squared, couldn't I, which is four.

And I could have two cubed, which is eight.

And I can have the number itself.

Oops, sorry.

Two cubed times five, which is 40.

I can have the five on its own.

I can have two times five, which is 10.

I can have two squared times five, which is 20.

Have I missed anything else? I think there's only one important one that we've missed out.

And it's often one that people forget.

And the reason that people forget it, and it's one, the reason that we forget it is because when we write out our prime factor form, write it as just as the primes, two to the power of three multiplied by five, technically we've got one lot of that, so people often forget one.

Now let's check that we got all the factors there, the factors of 40.

One and 40 we've got, two and 20 we've got, four and 10 we've got, and five and eight we've got.

So we can use the products of prime factors to find all the factors of a number.

Pause the video and have a go.

When you're ready unpause it and we can go through it together.

Good luck.

So how did you get on? I put the answers to question one on the screen now.

Let's go through question two together.

Number 30.

Well the factors, the prime factors, are two, three, and five.

Now we could also do two times three, we could do two times five, and we could do two times three times five.

We can also do three times five.

Factor 30, and we have all of these numbers worked out.

That's one, two, three, five, six, 10, 15, and 30.

That's eight factors of 30.

Let's have a different colour.

Now let's do 42.

Now 42, well that's made up of two times three times seven.

So we've got two as a prime factor.

We could do two times three, which is six, we could do two times seven, which is 14, and we could do it two times three times seven, which is 42.

We could also do a three times seven for 21.

And that's going to be all the factors of 42.

The factors of 42 are one, two, three, six, seven, 21, 42.

You know what, I missed that 14, haven't I? Then we go, we've got eight factors again.

Now let's look at 20.

So 20.

Once again, two times two times five.

So I can write down on the two and I can write down the five, but I can't put another two in this column here because I've already used two.

Now I can have two times two.

Nothing wrong with that.

I can up two times five.

And I can have two times two times five.

But in this case, even though there are three prime factors, because one of the numbers repeats there aren't as many factors, and that's actually what's going to come up in question three.

What's the same? Well in each case they've each got three prime factors, but what's different is the number of factors.

And because with 20 one of the prime factors repeats it means that we don't have as many factors of 20 as you do of 30, 42, and 70.

Just write these out.

So I've got one, two, four, five, 10, and 20.

Sorry, I answered question three before I finished question two.

Finally, 70.

It's made up of two multiplied by five multiplied by seven.

So I could also have two times five, and I can up two times seven, and I can up two times five times seven.

I can also have five times seven.

So when I factor the 70, a one, two, seven, 10, 14.

Missed out five, didn't I? Five, 35, and 70.

So once again, I've got eight factors, because the number didn't repeat, did they? We'll finish with the explore activity.

I've got a number in the middle, two to the power of four multiplied by three squared multiplied by five squared.

And I've written four statements around the side that are either true or false.

Your task is to work out which ones are true and which ones are false and try and add some more statements to the tree diagram.

Pause the video and have a go.

When you're ready unpause it and we'll go through it together.

You can pause in three, two, one.

Okay, how did you get on? I'm going to start with saying 30 is a factor.

Is that true or false? Well 30 would be five times six and six is two times three.

So we've got two and we've got a three and we've got a five in there, so yes, 30 is a factor, that's true.

The last digit is a zero.

If the last digit is a zero then the number must be in the 10 times table, it must be a multiple of 10, which means that we must be able to make 10 with some of our prime factors.

And we've got a five and we've got a two so that one's also true.

Now, if the last digit's zero then it's even, which means that it's definitely not an odd number.

And actually, anytime you multiply a number by two it will always be even.

So if I wrote up this number here, three to the power of four times five to the power of three times seven to the power of nine times two, you can tell it me it will definitely be even, I'm timesing it by two.

Even if I times it by two to the power of three, I'm just timesing it by two three times, definitely even.

Is it a square number? And if it's a square number that means that we can find a number that when multiplied by itself gives us that value.

Now we can break that down.

Two to the power of two times three times five and two to the power of two times three times five.

If I were to multiply these together I'll get the original number, won't I? And that means that I've got something multiplied by itself.

So that's true.

Whatever this number is, it is a square number.

Did you come up with anything else? If so, we'd love to see it.