Greatest common factor examples

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, or greatest common divisor, of 20 of 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 way 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-- factor of both 20 and 40 is actually 20. 20 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 now 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 GCD of 10 and 7? And there's two ways that you can approach this. One way, you could literally list all of the factors-- not prime factors, just regular factors-- of each of these numbers and figure out which one is greater or what is the largest factor of both. So, for example, you could say, well, I got a 10, and 10 can be expressed 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. Seven-- what are all of its factors? Well, 7 is prime. It only has two 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 factor. So the greatest common factor of 10 and 7, or the greatest common divisor, is going to be equal to 1. So let's write that down. 1. Let's do one more. What is the greatest common divisor of 21 and 30? And this is 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 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 the factors. Let me do it that way really fast. So if I say 21, what are all the factors? Well, it's 1 and 21, and 3, and 7. I think I've got all of them. And 30 can be written as 1 and 30, 2 and 15, and 3-- actually, I'm going to run out of them. Let me write it this way so I get a little more space. So 1 and 30. 2 and 15. 3 and 10. And 5 and 6. So here are 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 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 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, the only thing that's common right over here is a 3. And so we would say that 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 say the greatest common divisor is one. Let me give you another interesting example, just so that we can get a sense of things. So let's say these two numbers were not 21 and 30, but let's say we care about the greatest common divisor not of 21, but let's say of 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 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, 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 could 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 into both numbers.