0:00:00.860,0:00:03.840
We're asked, what is the[br]greatest common divisor

0:00:03.840,0:00:04.679
of 20 and 40?

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And they just say,[br]another way to say

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this is the GCD, or greatest[br]common divisor, of 20 of 40

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is equal to question mark.

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And greatest common divisor[br]sounds like a very fancy term,

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but it's really[br]just saying, what

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is the largest number that is[br]divisible into both 20 and 40?

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Well, this seems like a pretty[br]straightforward situation,

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because 20 is actually[br]divisible into 40.

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Or another way to[br]say it is 40 can

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be divided by 20[br]without a remainder.

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So the largest[br]number that is a-- I

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guess you could say-- factor of[br]both 20 and 40 is actually 20.

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20 is 20 times 1,[br]and 40 is 20 times 2.

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So in this situation,[br]we don't even

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have to break out our paper.

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We can just write 20.

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Let's do a couple more of these.

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So we're asked, what is[br]the greatest common divisor

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of 10 and 7?

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So let's now break out[br]our paper for this.

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So our greatest common[br]divisor of 10 and 7.

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So let me write that down.

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So we have 10.

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We want to think about what[br]is our GCD of 10 and 7?

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And there's two ways that[br]you can approach this.

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One way, you could literally[br]list all of the factors--

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not prime factors, just[br]regular factors-- of each

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of these numbers and figure[br]out which one is greater

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or what is the largest[br]factor of both.

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So, for example, you could[br]say, well, I got a 10,

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and 10 can be expressed[br]1 times 10 or 2 times 5.

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1, 2, 5, and 10.

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These are all factors of 10.

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These are all, we could[br]say, divisors of 10.

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And sometimes this is called[br]greatest common factor.

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Seven-- what are[br]all of its factors?

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Well, 7 is prime.

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It only has two[br]factors-- 1 and itself.

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So what is the[br]greatest common factor?

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Well, there's only one[br]common factor here, 1.

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1 is the only common factor.

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So the greatest common[br]factor of 10 and 7,

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or the greatest common divisor,[br]is going to be equal to 1.

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So let's write that down.

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1.

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Let's do one more.

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What is the greatest common[br]divisor of 21 and 30?

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And this is just another[br]way of saying that.

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So 21 and 30 are the two[br]numbers that we care about.

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So we want to figure out[br]the greatest common divisor,

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and I could have written[br]greatest common factor,

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of 21 and 30.

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So once again, there's[br]two ways of doing this.

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And so there's the way I did[br]the last time where I literally

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list all the factors.

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Let me do it that[br]way really fast.

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So if I say 21, what[br]are all the factors?

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Well, it's 1 and[br]21, and 3, and 7.

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I think I've got all of them.

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And 30 can be written as 1 and[br]30, 2 and 15, and 3-- actually,

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I'm going to run out of them.

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Let me write it this way so[br]I get a little more space.

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So 1 and 30.

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2 and 15.

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3 and 10.

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And 5 and 6.

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So here are all of[br]the factors of 30.

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And now what are[br]the common factors?

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Well, 1 is a common factor.

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3 is also a common factor.

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But what is the[br]greatest common factor

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or the greatest common divisor?

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Well, it is going to be 3.

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So we could write 3 here.

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Now, I keep talking[br]about another technique.

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Let me show you the[br]other technique,

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and that involves the[br]prime factorization.

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So if you say the prime[br]factorization of 21-- well,

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let's see, it's divisible by 3.

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It is 3 times 7.

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And the prime factorization[br]of 30 is equal to 3

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times 10, and 10 is 2 times 5.

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So what are the[br]most factors that we

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can take from both 21 and[br]30 to make the largest

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possible numbers?

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So when you look at the[br]prime factorization,

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the only thing that's common[br]right over here is a 3.

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And so we would say that[br]the greatest common factor

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or the greatest common[br]divisor of 21 and 30 is 3.

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If you saw nothing in[br]common right over here,

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then you say the greatest[br]common divisor is one.

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Let me give you another[br]interesting example, just so

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that we can get a[br]sense of things.

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So let's say these two[br]numbers were not 21 and 30,

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but let's say we care about[br]the greatest common divisor not

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of 21, but let's[br]say of 105 and 30.

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So if we did the prime[br]factorization method,

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it might become a[br]little clearer now.

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Actually figuring out, hey,[br]what are all the factors of 105

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might be a little bit[br]of a pain, but if you

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do a prime factorization,[br]you'd say, well, let's see,

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105-- it's divisible[br]by 5, definitely.

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So it's 5 times 21,[br]and 21 is 3 times 7.

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So the prime[br]factorization of 105

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is equal to-- if I write them[br]in increasing order-- 3 times

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5 times 7.

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The prime factorization of[br]30, we already figured out

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is 30 is equal to[br]2 times 3 times 5.

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So what's the most[br]number of factors

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or prime factors that[br]they have in common?

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Well, these two both have a[br]3, and they both have a 5.

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So the greatest common factor[br]or greatest common divisor

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is going to be a[br]product of these two.

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In this situation,[br]the GCD of 105 and 30

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is 3 times 5, is equal to 15.

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So you could do it either way.

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You could just list out the[br]traditional divisors or factors

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and, say, figure[br]out which of those

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is common and is the greatest.

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Or you can break it down[br]into its core constituencies,

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its prime factors,[br]and then figure out

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what is the largest set[br]of common prime factors,

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and the product[br]of those is going

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to be your greatest[br]common factor.

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It's the largest number that[br]is divisible into both numbers.