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