Skip to 0 minutes and 7 seconds MICHAEL ANDERSON: We can take any number and we can split it down into a prime number that multiply together to get that number. So let’s take a look at say 68.

Skip to 0 minutes and 19 seconds PAULA KELLY: So we’re looking for some prime numbers that are multiplied together to give us 68.

Skip to 0 minutes and 24 seconds MICHAEL ANDERSON: Yeah. There are lots of ways of doing this, but we’re going to use prime factor trees. So we’re going to look for two numbers that multiply together to give us our 68. So 68 is even, so we could do 2, and we’re going to multiply that by 34.

Skip to 0 minutes and 42 seconds PAULA KELLY: Because 2 times 34 is 68. OK.

Skip to 0 minutes and 45 seconds MICHAEL ANDERSON: Now 2 is a prime numbers. I’m going to circle that. That’s going to be part of our final multiplication, but 34 isn’t prime. It’s an even number. I can break this one down into two multiplied by 17.

Skip to 1 minute and 0 seconds PAULA KELLY: So you circled 17, because 17 is prime? OK, 17 is also a factor of 68. Could you start it by dividing by 17 to begin with?

Skip to 1 minute and 11 seconds MICHAEL ANDERSON: Yeah, 17 multiplied by 4 would give a 68, and we could split that 4 up into 2 times 2. So whichever way you choose to break this down, you’ll always end up with the same prime numbers. So in this case, 68 is equal to 2 multiplied by 2, multiplied by 17. And we can write this a little bit shorter by saying that’s 2 to the power of 2, 2 squared, multiplied by 17.

Skip to 1 minute and 39 seconds PAULA KELLY: OK, so we’ve seen how we could write the number 68 as a product of primes as our prime numbers multiply together. So would that same method work any number?

Skip to 1 minute and 49 seconds MICHAEL ANDERSON: Works for every single number, but just the larger the number, the more primes are going to be involved. So let’s have a look at say 3,150.

Skip to 1 minute and 59 seconds PAULA KELLY: That’s a quite large number, good, OK.

Skip to 2 minutes and 2 seconds MICHAEL ANDERSON: And we’re looking for two numbers that multiply together to give us 3,150. So we could start by looking at maybe 315 multiplied by 10.

Skip to 2 minutes and 13 seconds PAULA KELLY: So they’re not both prime.

Skip to 2 minutes and 15 seconds MICHAEL ANDERSON: No, so with this prime factor tree, what we’re going to have to do is have more branches coming under the 315, and more branches going under the 10. OK, so let’s have a look at 315. Ends in a 5, so it’s going to be in the 5 times tables, and that’s going to be 5 multiplied by 63. Now we found our first prime, 5. 63 is in the 3 times tables. It’s three lots of 21. 3 is a prime number, but 21 isn’t. 21 is 3 multiplied by seven, two primes multiplied together.

Skip to 2 minutes and 52 seconds PAULA KELLY: So you’ve circled all your primes. So we know we finished this chain. 10 isn’t prime so–

Skip to 2 minutes and 58 seconds MICHAEL ANDERSON: Well let’s go back to that one. 10 is 2 multiplied by 5, and they’re both prime numbers so we’re finished. So what we can say is 3,150 is equal to, well let’s start with the smallest prime. We’ve got 2 multiplied by 3, multiplied by 3. We’ve got a 5, a 5, and a 7 left. 5, 5, and 7.

Skip to 3 minutes and 26 seconds PAULA KELLY: So that’s quite a long way of writing this. Is there a quicker way?

Skip to 3 minutes and 29 seconds MICHAEL ANDERSON: Yeah. So we got some multiples. We’ve got a 2. We’re going to multiply by two 3’s. So that’s 3 squared, multiplied by 5 squared, and then we’ve got an extra 7 there. So we could check. We could do 2 multiplied by 3 squared, multiplied by 5 squared, multiplied by 7, and that will give us 3,150.

Skip to 3 minutes and 53 seconds PAULA KELLY: OK so another number we could look at is 32. So this is quite interesting number and we see how we can write as a product and its prime factor.

Skip to 4 minutes and 3 seconds MICHAEL ANDERSON: OK, so we’ll draw on the prime factor tree. We’ve got 32 at the top. And it’s often good to start to see if we can divide these numbers by 2, then 3, then 5, the smaller primes. So let’s half it. 2 goes into 32 16 times. 2 multiplied by 16 gives us 32. 2 is prime but 16 isn’t, but we can half it again to give us 2 and 8. 2 is prime but 8 isn’t, we can half it again. Give us 2 and 4, and then we can split the four up into another 2 and a 2. So 32 is interesting because it’s only got 2 as a prime number factor.

Skip to 4 minutes and 47 seconds It is 2, times 2, times 2, times 2, times 2, and that gives you 32. So 32 can be written as one, two, three, four, five, 2 to the power of 5.

Skip to 4 minutes and 59 seconds PAULA KELLY: So 32 written as a product of primes, and it with a power is multiplied by itself. So 2 multiplied by itself five times will give us 32.

# Prime factorisation

In the previous step we learnt that the definition of a prime number is a positive integer which has exactly two distinct factors. These two factors will always be the number 1 and the number itself.

The fundamental theorem of arithmetic states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that this representation is unique. This is sometimes referred to as the “unique factorisation” theorem.

In this video Michael and Paula demonstrate how prime factor trees can be used to help express a number as a product of its prime factors.

## Suggest

In the video Michael says that the larger the number the more prime factors it will have. In many cases this is true, but not always.

Can you think of a large number which will have a small number of prime factors? Describe the method you use to find your number.

## Problem worksheet

Now complete questions 8 and 9 from this week’s worksheet.

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