in this video we're going to look at a process called prime factor decomposition and this is sometimes called writing a number as a product of its prime factors we're going to begin by reminding ourselves what we mean by some of the key terms the first term we're interested in is the word factor and that's a number that divides exactly into another without leaving a remainder then we have prime numbers those are positive integers positive whole numbers which have exactly two factors one in itself we're going to need the first few prime numbers for this video they are 2 3 5 7 11 and 13 see if you can memorize them now pause the video and give yourself a minute try and remember as many as you can okay great so the process we're looking at is called prime factor decomposition and we sometimes call it writing a number as a product of its prime factors now it's helpful to remember that the word product means the result we get when we multiply two numbers together and when we write a number as a product of its prime factors we're trying to find all the prime numbers which multiply together to make that original number we can use a prime factor tree to do this or just the list let's see what this looks like our first example is to write the number 40 as the product of its prime factors now my preferred method is to use something called a factor tree and what we do is we start with the number 40 at the top of the tree and we then look for two factions of 40 specifically a factor pair and if one of those factors is a prime number even better well forty can be written as 2 times 20 now 2 is a prime number and so we circle the two and this means we stop here at this branch our next job is to repeat this but this time we do it with the number 20 what 20 is 2 times 10 and 2 as we saw is a prime number so we can circle this 2 and we stop this branch here lest it is again for the number 10 let's write 10 as 2 times 5 well we know 2 is a prime number so we're going to circle it but we also know that 5 is a prime number so we circle 5 2 now of course when we circle a number we stop that branch so we can't go any further now a common misconception here is to think that simply by drawing the factor tree that we're finished and another mistake is to write these as a list with commas or as a thumb with addition signs but remember the process is called writing a number as a product of its prime factors where product means x so we have to write these using multiplication signs we take each prime number that we've circled it's 2 times 2 times 2 times 5 now we could write this in index form as 2 cubed times 5 but the question hasn't specified here so either form is absolutely fine note that we don't actually need to work this out if we were to work out it would just give us the original number of 40 and you might also have noticed that we needn't have started with the factor pair 2 and 20 we could have used 4 and 10 and the same result so 4 times 10 is 40 then 4 is 2 times 2 and we circle our twos similarly 10 is 2 times 5 then we circle the 2 and the 5 and once again we've got 2 times 2 times 2 times 5 let's have a look at another example this time we're going to write 120 as the product of its prime factors but the question has specified to give our answer in index form we begin of course with 120 at the top of our tree and we split into a factor pair now I see that 120 is an even number and so it's divisible by 2 and I know that 2 is prime so I'm going to do 2 times 60 as my first factor pair and then I'm going to circle the number two since its prime since I've popped a circle around the number 2 I'm done with this branch and so I move on to the number 60 once again 60 is an even number and so it's divisible by 2 and in fact is the result of 2 times 30 so I circle the 2 and I stop that branch there we now move on to 30 30 is still even so I write it as 2 times 15 and pop a circle around the 2 and then I go back to the 15 now 15 is not even so I'm going to go up to the next prime number which is 3 and I know that 15 is 3 times 5 well in fact 5 is also a prime number so I can circle both branches and I stopped here of course this doesn't mean I'm finished I need to take all my prime numbers and I need to write them as a product where product means times and so I can say that 120 is equal to 2 times 2 times 2 times 3 times 5 but I'm still not finished I need to write my answer in index form which means I need to use powers or indices and so I write this as 2 cubed times 3 times 5 and now we're finished 120 as the product of its prime factors is 2 cubed times three times five so now that we have the basic concept let's look at a few examples now it's your turn to have a go make your way over to the activity back called B prime factors worksheet pack now on this pack there are a lot of numbers for you to attempt to write as a product of their prime factors I wouldn't do them all I would say four or so from each list should be enough there's also a challenge at the bottom of this page pause the video work your way through the prime factors worksheet back give yourself a challenge if you feel like it and then come back and we'll work through the challenge together so let's have a look at the challenge we have a number P which written is its product of its prime factors is two times three squared times a and we want to find the value of a or a value of a that makes P a square number so P is two times three squared times a we might notice that 3 squared is itself a square number it's the product of 3 times 3 which is 9 we have 2 so what could a be so that P itself is a square number well since we've already got 3 squared if we can make 2 squared then P will be 2 squared times 3 squared so let's try letting a be equal to 2 then P is equal to 2 times 3 squared times 2 which is 2 squared times 3 squared so how do we know that this is a square number well we can write it as 2 times 3 all squared this is the same thing or 6 squared and by definition this absolutely must be a square number so a could be equal to 2 but we're there any other options well in fact there were what about if we let a be equal to 8 then P is 2 times 3 squared times 8 but of course 8 is itself 2 cubes so this becomes 2 to the fourth power times 3 squared which we can then write as 2 squared times 3 all squared and once again by definition that has to be a square number one so two of the values we could have chosen for a are 2 and 8 of course there are others but these are the two smallest values of a we could have chosen and so we now concluded this video we've learned how to write a number as a product of its prime factors by using a factor tree thank you for watching and hopefully we'll see you back here soon