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CCSS.Math:

we're asked what is the greatest common divisor of 20 and 40 and they just say another way to say this is the GCD your greatest common divisor of 20 and 40 is equal to question mark and greatest common divisor sounds like a very fancy term but it's really just saying what is the largest number that is divisible into both 20 and 40 well this seems like a pretty straightforward situation because 20 is actually divisible into 40 or another ways to say it is 40 can be divided by 20 without a remainder so the largest number that is a I guess you could say a factor of both 20 and 40 is actually 20 20 is 20 divided is 20 times 1 and 40 is 20 times 2 so in this situation we don't even have to break out our paper we can just write 20 let's do a couple more of these so we're asked what is the greatest common divisor of 10 and 7 so let's not break out our paper for this so our greatest common divisor of 10 and 7 so let me write that down so we have 10 we want to think about what is our G C D of of 10 and 7 and there's two ways that you can approach this one way you can literally list all of the factors not prime factors just regular factors of each of these numbers and figure out which one is greater so or what is the largest factor of both so for example you could say well I got a 10 I got a 10 and 10 can be expressed 10 can be expressed as 2 or let me be careful as 1 times 10 or 2 times 5 1 2 5 and 10 these are all factors of 10 these are all we could say divisors of 10 and sometimes this is called greatest common factor 7 what are all of its factors well 7 is prime it only has 2 factors 1 and itself so what is the greatest common factor well there's only one common factor here 1 1 is the only common so the greatest common factor of ten and seven or the greatest common divisor is going to be equal to equal to one so let's write that down one let's do one more what is the greatest common divisor of 21 and 30 and they just just just another way of saying that so 21 and 30 are the two numbers that we care about so we want to figure out the greatest common divisor and I could have written greatest common factor of 21 21 and 21 and 30 so once again there's two ways of doing this and so there's the way I did the last time where I literally list all of the factors let me do that way really fast so if I say 21 what are all the factors well it's 1 and 21 1 and 21 and 3 and 7 and I think I've got all of them and 30 30 can be written as 1 + 32 + 15 + 3 actually I'm going to run out of them let me write it this way so get a little more space 1 + 32 + 15 3 + 10 + 5 + 6 5 + 6 so here are all of the factors all of the factors of 30 and now what are the common factors well 1 is a common factor 3 is also a common factor but what is the greatest common factor or the greatest common divisor well it is going to be it is going to be 3 so we could write 3 here now I keep talking about another technique let me show you the other technique and that involves the prime factorization so if you say the prime factorization of 21 well let's see it's divisible by 3 it is 3 times 7 and the prime factorization of 30 is equal to is equal to let's see it's 3 times 10 and 10 is 2 times 5 so what are the most factors that we can take from both 21 and 30 to make the largest possible numbers so when you look at the prime factorization prime factorization the only thing that's common the only thing that's common right over here is a 3 and so we would say that the greatest common factor the greatest common factor or the greatest common divisor of 21 and 30 is 3 if you saw nothing in common right over here then you would say the greatest common divisor is 1 let me give you another interesting example just so that we can get a sense of things so let's say that we had let's say these two numbers were not 21 and 30 but let's say we care about the greatest common divisor greatest common divisor not of 21 but let's say of 105 105 and 30 on 105 and 30 so if we did the prime factorization method it might become a little clearer now actually figuring out hey what are all the factors of 105 might be a little bit of a pain but if you do a prime factorization you'd say well let's see 105 it's divisible by 5 definitely so it's 5 times 21 and 21 is 3 times 7 so the prime factorization of 105 is equal to if I write them in increasing order 3 times 5 times 7 the prime factorization of 30 we already figured out we already figured out is 30 is equal to 2 times 3 times 5 so what's the most number of factors or prime factors that they have in common well these two both have a 3 they both have a 3 and they both have a 5 so the greatest common factor or greatest common divisor is going to be a product of these two in this situation the GCD of 105 and 30 is 3 times 5 is equal to 15 so you could do it either way you can just list out the traditional divisors or factors and say figure out which of those is common and is the greatest or you can break it down into its core constituencies its prime factors and then figure out what is the largest set of common prime factors and the product of those is going to be your greatest common factor it's the largest number that is divisible in to both numbers