How to Find the GCF and LCM of 3 Numbers Using Prime Factorization | Math with Mr. J

welcome to math with Mr J [Music] this video I'm going to cover how to find the greatest common factor and least common multiple of three numbers using prime factorization now I like using prime factorization and find it helpful when working with numbers that are a little larger in value and not as simple to work with for example the strategy of listing out all of the factors or listing some multiples of the numbers you're working with can be kind of difficult and time consuming when working with larger numbers in value so prime factorization is a different approach a different strategy to be familiar with when it comes to finding the greatest common factor or least common multiple let's start with greatest common factor and jump into our example where we have 54 36 and 90. so we need to find the greatest common factor of those three numbers let's start with the prime factorization of all three of our numbers starting with the prime factorization of 54. 2 times 27 equals 54. so 2 and 27 are factors of 54. now 2 is prime so we are done there 27 we can break down three times nine equals 27. so 3 and 9 are factors of 27. 3 is prime so we are done there 9 we can break down three times three equals nine so three is a factor of nine three is prime so we are done there and there and that's the prime factorization of 54. we can't break that down any further let's move on to the prime factorization of 36 2 times 18 equals 36 so 2 and 18 are factors 2 is prime so we are done there we can break 18 down further two times nine equals eighteen so two and nine are factors 2 is prime so we are done there we can break 9 down further three times three equals nine so three is a factor of nine three is prime so we are done there and there and that's the prime factorization of 36. we can't break that down any further and then lastly we have the prime factorization of 90. 2 times 45 equals 90. so 2 and 45 are factors 2 is prime so we are done there 45 we can break down further five times nine equals 45 so 5 and 9 are factors 5 is prime so we are done there 9 we can break down three times three equals nine so three is a factor 3 is prime so we are done there and there and that's the prime factorization of 90. we can't break that down any further now that we have the prime factorization of all three of our numbers we need to look for common prime factors so prime factors that all three of our numbers share that they all have in common now in order to make it easier to find common prime factors I'm going to list the prime factors off to the side here starting with 54. so we have 2 3 3 and 3 2 times 3 times 3 times 3 equals 54. now let's move on to 36 we have 2 2 3 and 3 2 times 2 times 3 times 3 equals 36. and then lastly we have 90. 2 3 and 5 2 times 3 times 3 times 5 equals ninety now that we have the prime factors listed we need to look for common prime factors so factors that all three of these numbers share so two is a common prime factor three is a common prime factor and then we have another three in common the common prime factors are 2 3 and 3. now once we have those common prime factors we multiply them to get the greatest common factor two times three is six times three is eighteen so the GCF the greatest common factor of 54 36 and 90. is 18. so there's how to find the GCF of three numbers using prime factorization let's move on to the LCM here's our example for finding the least common multiple of three numbers using prime factorization let's jump into our example where we have 21 28 and 32. so we need to find the least common multiple of those three numbers let's start with the prime factorization of all three of our numbers starting with 21. foreign so three and seven are factors of 21. now 3 is prime so we are done there 7 is prime as well so we are done there and that's the prime factorization of 21. we can't break that down any further let's move on to 28. 2 times 14 equals 28 so 2 and 14 are factors of 28. 2 is prime so we are done there 14 we can break down 2 times 7 equals fourteen so two and seven are factors of 14. two is prime so we are done there 7 is prime as well so we are done there and that's the prime factorization of 28. we can't break that down any further lastly we have the prime factorization of 32. two times sixteen equals 32 so 2 and 16 are factors of 32. 2 is prime so we are done there 16 we can break down two times eight equals 16. so 2 and 8 are factors of 16. 2 is prime so we are done there we can break eight down further two times four equals eight so two and four are factors of eight two is prime so we are done there we can break four down further two times two equals four so two is a factor of four two is prime so we are done there and there and that's the prime factorization of 32. we can't break that down any further now that we have the prime factorization of all three of our numbers we need to list the prime factors and match them vertically let's start with the prime factors of 21. so 3 and seven three times seven equals 21. now for the prime factors of 28 and I'm going to start with the prime factor of 7 since 21 also has a prime factor of 7. I'm matching the sevens vertically then we have 2 and 2. so 7 times 2 times 2 equals 28 and then lastly we have the prime factors of 32. we have a 2 that we can match vertically another two that we can match vertically and then three more twos now that we have our prime factors listed and matched vertically we need to bring down a number to represent each column and it's going to be one for each column you'll see what this means as we go through this step so I like to draw a line to separate our steps here and now again we need to go through each column and bring down a number to represent that column for example we have a three column right here even though there's one three that's okay we have a three column there so let's bring down a 3 to represent that column times we have a 7 column here now there are two sevens there but we only bring down one seven to represent that column times next we have a column of twos so bring down a two to represent that column and then another column of twos so bring down another two to represent that column times we have one two here but it is a column so bring down a two times another 2 here times another two here so we end up with three times seven times two times two times two times two times two and by multiplying those numbers we get the least common multiple now there are different ways to go about multiplying these numbers for example you can start with three times seven and work your way from left to right I'm going to start with this group of five twos right here so we have two times two times two times two times two two to the fifth power we have two expanded out five times and multiplied so two times two is four times two is eight times two is sixteen times two is thirty-two then we can bring down our seven and three now again there are different ways to go about multiplying these numbers I'm just going to work from left to right so I'll do three times seven next which is twenty one and then we have 21 times 32 for our last step now I'm not sure what 21 times 32 is so I need to work this out off to the side so 32 times 21. I put the larger number in value on top let's multiply one times two is two one times three is three we are done with this one and now we need that zero two times two is four and then 2 times 3 is 6. now we add 32 plus 640 2 plus 0 is 2 3 plus 4 is 7. and then we have that six so six hundred seventy two and that's our least common multiple so I'll put an arrow there so 3 times 7 times 2 times 2 times 2 times 2 times 2 equals 672. so the LCM the least common multiple of 21 28 and 32 is 600 72. so there you have it there's how to find the GCF and LCM of three numbers using prime factorization I hope that helped thanks so much for watching until next time peace