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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

# 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