Lesson 16: Common factors
Welcome to the greatest common divisor or greatest common factor video. So just to be clear, first of all, when someone asks you whether what's the greatest common divisor of 12 and 8? Or they ask you what's the greatest common factor of 12 and 8? That's a c right there for common. I don't know why it came out like that. They're asking you the same thing. I mean, really a divisor is just a number that can divide into something, and a factor-- well, I think, that's also a number that can divide into something. So a divisor and a factor are kind of the same thing. So with that out of the way, let's figure out, what is the greatest common divisor or the greatest common factor of 12 and 8? Well, what we do is, it's pretty straightforward. First we just figure out the factors of each of the numbers. So first let's write all of the factors out of the number 12. Well, 1 is a factor, 2 goes into 12. 3 goes into 12. 4 goes into 12. 5 does not to go into 12. 6 goes into 12 because 2 times 6. And then, 12 goes into 12 of course. 1 times 12. So that's the factors of 12. Let's write the factors of 8. Well, 1 goes into 8. 2 goes into 8. 3 does not go into 8. 4 does go into 8. And then the last factor, pairing up with the 1 is 8. So now we've written all the factors of 12 and 8. So let's figure out what the common factors of 12 and 8 are. Well, they both have the common factor of 1. And that's really not so special. Pretty much every whole number or every integer has the common factor of 1. They both share the common factor 2 and they both share the common factor 4. So we're not just interested in finding a common factor, we're interested in finding the greatest common factor. So all the common factors are 1, 2 and 4. And what's the greatest of them? Well, that's pretty easy. It's 4. So the greatest common factor of 12 and 8 is 4. Let me write that down just for emphasis. Greatest common factor of 12 and 8 equals 4. And of course, we could have just as easily had said, the greatest common divisor of 12 and 8 equals 4. Sometimes it does things a little funny. Let's do another problem. What is the greatest common divisor of 25 and 20? Well, let's do it the same way. The factors of 25? Well, it's 1. 2 doesn't go into it. 3 doesn't go into it. 4 doesn't go into it. 5 does. It's actually 5 times 5. And then 25. It's interesting that this only has 3 factors. I'll leave you to think about why this number only has 3 factors and other numbers tend to have an even number of factors. And then now we do the factors of 20. Factors of 20 are 1, 2, 4, 5, 10, and 20. And if we just look at this by inspection we see, well, they both share 1, but that's nothing special. But they both have the common factor of? You got it-- 5. So the greatest common divisor or greatest common factor of 25 and 20- well, that equals 5. Let's do another problem. What is the greatest common factor of 5 and 12? Well, factors of 5? Pretty easy. 1 and 5. That's because it's a prime number. It has no factors other than 1 and itself. Then the factors of 12? 12 has a lot of factors. It's 1, 2, 3, 4, 6, and 12. So it really looks like only common factor they share is 1. So that was, I guess, in some ways kind of disappointing. So the greatest common factor of 5 and 12 is 1. And I'll throw out some terminology here for you. When two numbers have a greatest common factor of only 1, they're called relatively prime. And that kind of makes sense because a prime number is something that only has 1 and itself as a factor. And two relatively prime numbers are numbers that only have 1 as their greatest common factor. Hope I didn't confuse you. Let's do another problem. Let's do the greatest common divisor of 6 and 12. I know 12's coming up a lot. I'll try to be more creative when I think of my numbers. Well, the greatest common divisor of 6 and 12? Well, it's the factors of 6. Are 1, 2, 3, and 6. Factors of 12: 1, 2, 3-- we should have these memorized by now. 3, 4, 6, and 12. Well, it turns out 1 is a common factor of both. 2 is also a common factor of both. 3 is a common factor of both. And 6 is a common factor of both. And of course, what's the greatest common factor? Well, it's 6. And that's interesting. So in this situation the greatest common divisor-- and I apologize that I keep switching between divisor and factor. The mathematics community should settle on one of the two. The greatest common divisor of 6 and 12 equals 6. So it actually equals one of the numbers. And that makes a lot of sense because 6 actually is divisible into 12. Well, that's it for now. Hopefully you're ready to do the greatest common divisor or factor problems. I think I might make another module in the near future that'll give you more example problems.