Greatest Common Factor and Least Common Multiple | GCF and LCM | Math with Mr. J

welcome to math with Mr J [Music] in this video I'm going to cover greatest common factor and least common multiple both of these show up all throughout math everything from working with fractions to algebra to whatever else the case may be so no matter what level of math you're in or goal you are working towards these examples will help you better understand greatest common factor and least common multiple also known as GCF and LCM let's start with greatest common factor now as far as the greatest common factor between numbers this is going to be the largest factor in value that both numbers share factors are the numbers that can be multiplied to equal a given number or you can think of factors as the numbers a given number can be divided by evenly so divided with no remainder now it's not the most technical or mathematical way to word it but you can think of factors as the numbers that go into the given number so again the greatest common factor is going to be the greatest or largest factor in value that both numbers share that both numbers have in common let's jump into our examples and see exactly what this looks like starting with number one where we have 16 and 24. we first need to list the factors of each number let's start with the factors of 16. now I would suggest always starting with the factors of 1 and the number itself because we know one times that number will equal that number itself we can think of factors in terms of pairs 1 times 16 equals 16 or we can think of it in terms of division we can divide 16 by those factors evenly however you want to think about it so I'm going to write 1 and 16 with a gap in between for the other factors that way we can write the factors in order now we need to list the other factors of 16. so let's think about what else goes into 16 so to speak think about multiplication facts division facts and we can work our way up from one so there are different strategies and ways to work through this the next factors of 16 are 2 and eight two times eight equals sixteen so two and eight are factors of 16 and then our last factor of 16 is 4. 4 times 4 equals 16. so 4 is a factor of 16. although four times four equals 16 we only need to write 4 Once in our factors list and that's it for the factors of 16. you can always think about other possible factors if you're unsure if you're done for example 3 5 6 and so on but we have them all the factors of 16 are 1 2 4 8 and 16. let's move on to the factors of 24. we will start with 1 and 24. next we have two and twelve two times twelve equals 24. so 2 and 12 are factors of 24. then we have 3 and 8 3 times 8 equals 24 so 3 and 8 are factors of 24 and then lastly we have four and six four times six equals 24 so 4 and 6 are factors of 24 and that's it for the factors of 24 1 2 3 4 6 8 12 and 24. now that we have the factors listed for our numbers we need to look for common factors and specifically the greatest common factor so let's look for factors that these numbers share that they have in common so one is a common factor two is a common factor four is a common factor and then eight is a common factor so the greatest common factor is eight the greatest factor in value that they share the greatest factor in value that they have in common so the GCF the greatest common factor is eight the greatest common factor of 16 and 24 is eight let's move on to number two where we have 15 and 35. let's start with the factors of 15. we have one and fifteen next we have three and five three times five equals fifteen so three and five are factors of 15. and that's it for the factors of 15. one three five and fifteen now let's list the factors of 35 so we have one and 35 and then 5 and 7 5 times 7 equals 35 so 5 and 7 are factors of 35 and that's it for the factors of 35 1 5 7 and 35. now that we have our factors list for our numbers we need to look for common factors and specifically the greatest common factor so 1 is a common factor and then 5 is a common factor that means 5 is the greatest common factor so the GCF the greatest common factor is five the greatest common factor of 15 and 35 is 5. so there's how we find the greatest common factor let's move on to least common multiple here are our examples for least common multiple now as far as the least common multiple between numbers this is going to be the smallest multiple in value that both numbers share now a multiple is the result of multiplying a given number by an integer when we think of the multiples of a number we need to think about the numbers we get when multiplying that given number by integers a simpler way to think about multiples is to think about skip counting so all of the numbers something is going to hit when you count up by that number those are all going to be multiples this will make a lot more sense as we go through our examples let's jump into our examples starting with number one where we have 9 and 12. we're going to start by listing some multiples of both 9 and 12. then we will look for common multiples and specifically the least common multiple also referred to as the LCM let's start with some multiples of nine which are nine times one which is nine nine times two is eighteen nine times three is twenty-seven nine times four is 36 and 9 times 5 is 45 so you can see that we just skip counted by 9 to list those multiples 9 18 27 36 45 so on and so forth now I stopped at 45 because multiples go on forever they are endless they are infinite my suggestion is to list four or five multiples when looking for the least common multiple so list four or five multiples for each number look for any in common and if you don't have any in common you can always extend the multiples lists now let's list the first five multiples of twelve twelve times one is twelve twelve times two is twenty four twelve times three is thirty-six twelve times four is forty eight and twelve times five is sixty so again you can see that we skip counted there we skip counted by 12. so 12 24 36 48 60 so on and so forth now that we have some multiples listed for both 9 and 12 we need to look for any common multiples so any multiples that they share and then specifically we need to look for the least common multiple well 36 is a common multiple and it's going to be the least common multiple so the smallest multiple in value that they share so let's write that the LCM which stands for least common multiple is 36. so the least common multiple of 9 and 12 is 36. now one thing I do want to mention about common multiples is that they are infinite although we only have one common multiple in our lists as is 36 we can always extend multiples lists so we can always keep going to find more common multiples remember multiples are endless so that means common multiples are endless so that's just something to think about when it comes to multiples let's move on to number two where we have 10 and 25. let's start with some multiples of 10 so 10 times 1 is 10. 10 times 2 is 20 10 times 3 is 30 10 times 4 is 40. and then 10 times 5 is 50. so 10 20 30 40 50 so on and so forth now let's list the first five multiples of 25 so 25 times 1 is 25 25 times 2 is 50 25 times 3 is 75 25 times 4 is 100 and then 25 times 5 is 125. now that we have some multiples listed we can look for common multiples and specifically the least common multiple well 50 is a common multiple and it happens to be the least common multiple so the LCM is 50. the least common multiple of 10 and 25 is 50. so there you have it there's how to find the GCF and LCM I hope that helped thanks so much for watching until next time peace thank you