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  • 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.

Welcome to the greatest common divisor or greatest

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最大公因數解釋|因數和倍數|可汗學院|代數預備班 (Greatest common factor explained | Factors and multiples | Pre-Algebra | Khan Academy)

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    何虹達 發佈於 2021 年 01 月 14 日
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