this question says find the hcf which means the highest common factor and the lcm the lowest common multiple of 42 and 56. to do this we're not going to write down all the factors of both of them and all the multiples of both of them we're going to break them down into their prime factors first to make the working out a bit easier so i'm going to break both of these numbers down into their prime factors so i'll start with 42 so two numbers that multiply to make 42 i'll use 2 and 21 so 2 is a prime number 21 isn't 21 is 3 times 7 and 3 and 7 are both prime so 42 is 2 times 3 times 7 and 56 we can say that's two times twenty-eight twenty-eight is two [Music] fourteens and fourteen is two sevens so 56 can be rewritten as two times two times two times seven so now i've got the prime factors of 42 and the prime factors of 56 i'm gonna put them into a venn diagram so 42 and 56 so the factors they've both got i'm gonna put in the middle so they've both got a two and they've both got a seven 42 has also got a three so the 42 circle is two three and seven and the 56 circle needs two more twos so the 56 circle is two times two times two times seven so the highest common factor is the middle bit so the middle bit is two and seven which means two times seven so the highest common factor is two times seven which is fourteen the lowest common multiple is all of these numbers times together so 2 times 2 times 2 times 7 times 3 but this circle here remember is 56 so 2 times 2 times 2 times 7 is 56 so i can say the lowest common multiple is this circle 56 times 3. fifty times three is a hundred and fifty six times three is eighteen a hundred and fifty plus eighteen is a hundred and sixty eight so the lowest common multiple of 42 and 56 is 168. okay so there's a question for you to try here find the htf an lcm of 60 and 72. so we're going to start by breaking them down into their prime factors so for 60 i can say that's 2 times 30 and 2 is prime 30 i can break down more so 30 is 2 times 15 again 2 is prime and 15 can be broken down into 3 times 5. so 60 is 2 times 2 times 3 times 5 and 72 is two times thirty six thirty six is two eighteens eighteen is two nines and nine is three times three so 72 is two times two times two times three times three so i'm going to put this into a venn diagram again so i've got a 60 circle and a 72 circle so the factors the prime factors they've got in common go in the middle so they've both got two twos and a three so my sixty circle has got two twos of three and it needs a five and my seventy 72 circle has got two twos of three and it needs another two and another three so there's my venn diagram done the highest common factor are the numbers in the middle multiplied together so two times two times three two times two is four four times three is 12. and the lowest common multiple so i've got my 72 circle so 2 times 2 times 2 times 3 times 3 makes 72 multiplied by five so 72 multiplied by 10 is 720 half of that 360. here we've got a common exam type question so it says light a flashes every 12 seconds light b flashes every 15 seconds both lights flash at the same time work out how long it will take for both lights to flash at the same time again so what the question's asking us for is the lowest common multiple of 12 and 15. so we could write down the multiples of 12 and the multiples of 15 and find the lowest one or we could break them down into their prime factors and do the venn diagram again so i'm going to do the prime factors so i'm going to say 12 is two times six and six is two times three so twelve is equal to two times two times three fifteen is three times five and they are both prime numbers so 15 is three times five i'm going to put this into a venn diagram so i'm going to have a 12 circle and a 15 circle so they've both got three as a factor my 12 circle needs two twos and my 15 circle needs a five so the lowest common multiple is well it's 12 times five my 12 circle times five twelve times five is sixty so how long does it take for both lights to flash at the same time again that's 60 seconds okay so there's a question for you to try pause the video give it a go and press play when you're ready to go through the answer okay so we've got bus a leaves every 10 minutes bus b leaves every 14 minutes both buses leave at 9am work out the time both buses leave at the same time again so we can break these down into their prime factors and well we can say that 10 is equal to two times five and fourteen is equal to two times seven so if we have a ten circle and a 14 circle 2 is going to go in the middle that's the only common factor and then the 10 circle will have a 5 and the 14 circle will have a seven so the lowest common multiple is going to be so my 14 circle times five fourteen times five and fourteen times five well fourteen times ten is a hundred and forty half of that seventy so they both leave at the same time again in 70 minutes so what time will it be in 70 minutes so 60 minutes is till 10 am another 10 minutes so it's 10 10 am 10 past 10 here's a question with three numbers so find the highest common factor and the lowest common multiple of 8 12 and 18. so we're going to do the same thing again we're just going to have a venn diagram with three circles so we'll start with eight which is two times four two is prime and four is two times two so eight is two times two times two twelve is two times six and six is two [Music] times three so twelve is two times two times three and eighteen is two times nine and nine is three times three so eighteen is two times three times three so let's make a venn diagram with three circles so we're gonna have an eighth circle a 12 circle and an 18 circle so in the middle we're going to have the factors that all three of them have got so they've all got one two they've all got one two so the two goes in the middle eight and twelve have both got another two so in the eighth and the twelve circle we've got another two and twelve and eighteen have both got a three so 12 and 18 share a three and then there's another two in the eighth circle nothing else for the 12 circle and a three for the 18 circle so the highest common factor there's only one number in the middle so it's just two and the lowest common multiple is so all of these numbers multiplied together so we've got our 18 circle which is two times three times three 18 circle multiplied by two multiplied by two so eighteen times two times two eighteen times two is thirty six 36 times 2 is 72 okay so here's a question for you to try find the highest common factor and the lowest common multiple of nine twelve and fifteen so nine is three times three twelve is two times six and six is two times three so twelve is two times two times three and fifteen is three times five so i'm going to put this into a venn diagram and we'll have a nine circle i have a 12 circle and we'll have a 15 circle so what they all have in them they all have a 3. so threes in the middle that's going to be the highest common factor if they share anything else no so in my nine circle i need the three in my 12 circle two twos and my 15 circle needs a five so my highest common factor what's in the middle of all of them that is three and the lowest common multiple all of these numbers times together so if i have my 12 circle times three times five so twelve times three times five twelve times three is thirty-six thirty-six times five is a hundred and eighty okay to finish up two questions so we've got question one find the highest common factor of 72 and 96 and question two the lowest common multiple of 28 and 35 okay so question one we're going to break these numbers down into their prime factors so 72 is 2 times 36 36 is two 18s 18 is two nines and nine is three threes so 72 is 2 times 2 times 2 times 3 times 3 96 that's 2 times 48 48 is 2 24's 24 is 2 12. 12 is two sixes and six is two threes so we've got two times two times two times two times two times three so we'll make our venn diagram so we've got 72 and 96 what they both have so one two three twos and a three seventy two also got a three ninety 96 has also got two more twos we're after the highest common factor so the highest common factor is the numbers in the middle multiplied together so two times two times two times three so two times two is four times two is eight eight times three 24 and question 2 find the lowest common multiple of 28 and 35 so 28 we've got two fourteens and fourteen is two sevens 35 is in the five times table five sevens and they're both prime so 28 is two times two times seven thirty-five is five times seven so i'm going to put this into a venn diagram so i've got a 28 circle and a 35 circle so they've both got a seven so seven is the common factor the only common factor then 28's got two twos and 35 has got a 5. so the lowest common multiple is all of these numbers multiplied together so it's my 28 circle multiplied by 5 so 28 times 10 is 280 half of that 140. 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