Learn Greatest Common Factor (GCF) & Least Common Multiple (LCM) - [7] - Free Educational videos for Students in K-12 | Lumos Learning

Learn Greatest Common Factor (GCF) & Least Common Multiple (LCM) - [7] - Free Educational videos for Students in k-12


Learn Greatest Common Factor (GCF) & Least Common Multiple (LCM) - [7] - By Math and Science



Transcript
00:00 Hello . Welcome back . The title of this lesson
00:02 is called factors and multiples of numbers . This is
00:06 part one . Specifically , we're going to be talking
00:08 about reviewing and understanding what's called the greatest common factor
00:12 of two numbers and also the least common multiple of
00:15 two numbers . So what we wanna do is understand
00:18 each of these concepts independently and solve a lot of
00:20 problems to make sure that we have our skills where
00:22 they need to be . Now , what we're going
00:24 to do is use the skill later down the road
00:27 . When we deal a little bit more with fractions
00:29 will be using the greatest common factor and the least
00:32 common multiple when we do addition of fractions and some
00:36 other things with fractions down the road . So this
00:38 is an important skill for things that we will be
00:40 doing later . I'll also say that especially the concept
00:44 of factors right I would say is actually more important
00:47 because we're going to be using the idea of a
00:49 factor pretty much forever in Math would use it when
00:52 we get into algebra and other classes will be using
00:56 factors a lot . So it's the same concept as
00:58 what we're learning here as well . So let's talk
01:00 about the greatest common factor of two numbers . So
01:04 you might see this on a , you know ,
01:06 on a question or something like that , you might
01:09 see it as the G c f , so g
01:12 uh , cf this stands for greatest common factor .
01:16 So what that actually means , I'm gonna give you
01:18 two numbers . What we need to do is find
01:21 what we call the factors of those two numbers .
01:23 The factors of numbers are going to be a long
01:26 list of numbers . And then once we have the
01:28 factors , we need to find the largest one ,
01:30 what's the greatest one ? That is common to both
01:33 lists . So we're going to write the factors now
01:36 and all we have to do is pick the biggest
01:37 number That's in the same in both lists . That's
01:40 all it is . The greatest factor that is common
01:42 to both . That's what we're doing . Greatest .
01:44 Common factor now . The first thing you need to
01:47 know is what is a factor Anyway . When you
01:49 hear the word factor in math , this is what
01:52 I want you to think about . A factor is
01:55 just the numbers that multiply together to give you the
01:58 number that you want that you're talking about . I'll
02:01 say that one more time . The factors of the
02:03 number are all of the numbers that could be multiplied
02:06 together to give you the number you care about .
02:08 So for instance , the number 10 , right ?
02:10 You know that one times 10 is 10 . So
02:13 one and 10 are factors because they can be multiplied
02:16 together to give us 10 . Two and five are
02:19 also factors of 10 ? Why ? Because two times
02:22 five is 10 . So , when we say factors
02:24 of numbers , all we want to figure out is
02:26 what are all of the numbers that could be multiplied
02:28 to give us the number that we care about .
02:31 Which in this case the example was 10 . But
02:33 we can we can do it for any number that
02:35 we want . All right . So let's talk about
02:37 the greatest common factor of the number 12 and the
02:43 number 15 . So , what we have to do
02:46 is find the factors of 12 and write them down
02:49 and find the factors of 15 and write them down
02:52 . And then once we have all of the list
02:53 of factors , we just picked the biggest one that
02:56 is common in both lists . So let's write that
02:59 down . Greatest common factor of 12 . So ,
03:03 what we say is we write down these are the
03:05 factors fucking right . The word factor factor of 12
03:11 . And underneath that , we're gonna write the factors
03:15 of 15 . And we're gonna write these list of
03:19 factors down . And then once we have the list
03:22 , we'll just pick the greatest one that's common to
03:24 both list . And that thing is going to be
03:26 what we call the greatest common factor . All right
03:28 . So , in order to figure out , uh
03:31 there's two ways to really think about it . The
03:32 factors of 12 . You need to think about what
03:35 could be multiplied together to give you 12 . Another
03:37 way of thinking about it is can the number that
03:40 you're considering be divided in 2 , 12 evenly .
03:43 So , let's talk about this . What about the
03:45 number one ? We're just gonna go in order .
03:47 Starting with the number one . Can the number one
03:50 be multiplied by something to give us 12 ? Well
03:52 , one times 12 is 12 . So one is
03:54 a factor . What about the number two ? Two
03:57 times something gives me 12 ? Yes . Two times
04:00 six is 12 . So two is a factor .
04:01 All right . What about the number three ? Well
04:04 , three times four is 12 . So three is
04:07 also a factor . So it's starting to look like
04:09 like every single number is a factor . That's not
04:11 really quite right . But let's keep going . What
04:13 about four ? Four times something can give me 12
04:16 ? Yes . Four times three is 12 . So
04:18 four has to be there . Now , here we
04:20 come up against the number five is five . A
04:23 factor of 12 . 5 times what is 12 ?
04:27 Well , there is nothing that five can be multiplied
04:30 by to give you 12 . So , it's not
04:31 a factor . You see . Only things that can
04:34 go into the factor list are the things that can
04:36 be multiplied to give you 12 . So let's skip
04:40 over five . What about six ? Well , six
04:42 times two is 12 . So six is a factor
04:44 What about seven ? No seven can't be multiplied to
04:47 give me 12 . What about eight ? No ,
04:49 eight times two is 16 . It doesn't give me
04:50 12 . What about nine , nope not is not
04:53 a factor 10 . 10 is not a factor 11
04:55 . 11 is not a factor I can't multiply them
04:57 to give me 12 . But when I get to
04:59 the number 12 itself , 12 times one is 12
05:03 . So actually 12 is also a factor . All
05:06 right . So , the factors of the number 12
05:08 are 12346 and 12 . Because every one of those
05:12 numbers can be multiplied by something to give me 12
05:15 . Another way of thinking about it is that all
05:18 of these numbers can be divided in evenly into 12
05:21 . You know , 12 divided by one . That
05:24 goes in a whole number of times 12 divided by
05:26 two would be six . It goes a whole number
05:28 of times 12 divided by 3 , 12 , divided
05:30 by 4 , 12 , divided by 6 , 12
05:33 , divided by 12 . They all go a whole
05:37 number of times . So they're all factors . So
05:39 you can think of factors as being they can be
05:41 multiplied together to give us the number or you can
05:43 think of factors as being able to be divided in
05:47 a whole number of times perfectly . And those are
05:50 also called the factors . Now we have to do
05:53 the same process for the number 15 . Well one
05:55 times 15 , 15 . So that's a factor Two
05:59 is to a factor no , because two can't be
06:01 multiplied by anything to give me 15 . What about
06:03 three ? Three times five is 15 ? So that
06:05 goes in , what about four ? Four is not
06:08 a factor , what about 55 times three is 15
06:11 , So five is a factor . What about six
06:13 , nope , Six can't be multiplied to give me
06:14 15 . What about 789 10 ? No , none
06:19 of those are factors 11 , 12 can't be multiplied
06:22 to give me this 13 , 14 . The only
06:25 other factor is the number 15 because 15 times one
06:28 is 15 . So another way of thinking about it
06:31 is one can divide into here . Three can divide
06:33 into here . Five can divide into here , 15
06:36 can divide into here , but no other numbers can
06:38 . So here are the factors of 12 and here
06:41 are the factors of 15 and we have all of
06:44 the factors written for both . All we have to
06:45 do now is just look in the list and figure
06:48 out what is the greatest common factor ? Well ,
06:51 one is a common factor , but it is not
06:55 the greatest common factor because we have we have to
06:58 continue looking to is in this list , but to
07:00 is not here . Three is in both . Then
07:03 we have five here , but five is not over
07:06 here . Four is here , but four is not
07:08 over here , 15 is here , but it's not
07:10 here , there's no other commonality , only one is
07:13 common and three is common and three is the greatest
07:17 common factor . It's the largest factor . So the
07:19 G c f is equal to three for these two
07:24 numbers , the greatest common factor is three . I
07:26 know that this process seems cumbersome and hard and weird
07:29 , but after a while you're going to get very
07:31 good at writing the factors down and then you're going
07:34 to be very good at figuring out the largest one
07:37 that's common , that's all you're doing . All right
07:40 . A couple things I want to say before we
07:41 move onto the next problem , notice that for the
07:44 factors of 12 We have to go through and figure
07:47 out all the factors . But notice the number one
07:49 is a factor and the number itself 12 is always
07:52 a factor . The reason number one , and the
07:55 number itself 12 in that case is a factor is
07:58 because one times 12 is 12 . Notice over here
08:01 , one was a factor of 15 . and 15
08:03 was also a factor of 15 . So the number
08:05 one and the number itself are always factors of the
08:08 number . So really the list , we have a
08:11 big list here , but really the number one and
08:13 the number itself is always a factor . The number
08:15 one and the number itself is always a factor .
08:18 All we have to do is find the other ones
08:19 that are in the middle . All right . And
08:21 the second thing I want to say is that we
08:24 could just go through this and find them all by
08:26 thinking about them because these numbers were pretty small .
08:29 But if I ask you the G c F of
08:32 a large number , it might be tough to find
08:34 all the factors . So what I want to do
08:36 is show you another way to find the factors that
08:39 we didn't need it for this problem . But it's
08:41 going to help us in a few problems when I
08:44 give you larger numbers . So I'm gonna show you
08:45 for an easier problem . So , you'll understand it's
08:48 actually kind of a fun process . What we're gonna
08:51 do is find what we call a factor tree .
08:54 Let's say . We want to find the factors of
08:56 12 Factors of 12 . All you do is draw
08:59 a little tree under here and think to yourself what
09:02 times what can give me 12 ? And you can
09:04 pick anything you want , you can pick two times
09:06 six to give you 12 Or you can pick three
09:08 times forward to give you 12 . You have total
09:10 freedom to do what you want . Let's pick three
09:12 Times four and put a little dot there . This
09:14 is telling me that three times four is giving me
09:16 12 now , over here , Under the four .
09:19 We try to split that up further , what times
09:21 what can give me for ? Well , two times
09:24 two is the thing that I think of that can
09:25 give me for . So what I've done is I've
09:28 split the 12 into three times four and I split
09:30 the four into two times two . Now you can
09:33 split the three into one times three , but notice
09:36 that it doesn't really get any simpler because one times
09:39 three they're both they're both they're called prime numbers .
09:42 We'll get into prime numbers later , but they're basically
09:45 can't be broken apart any further . Two is also
09:48 the simplest number you can get to as well .
09:50 You can put one times two down here . In
09:52 fact , I'll go ahead and do that in red
09:53 . You can put like for instance one times three
09:56 , but you can't go any farther because then you
09:58 just keep breaking the three until one times three over
10:00 and over again . And you can make this one
10:02 times two and you can make this one times two
10:04 . But notice you can't go any farther because you
10:06 keep getting twos . And so the tree is basically
10:09 done here in blue . Now , everything in this
10:12 tree is a factor of the number 12 . Notice
10:16 that the number one is here down here at the
10:18 bottom , that's a factor of 12 . The number
10:20 two is in this tree . Don't don't worry about
10:22 the fact that it's in here a bunch of different
10:24 times . You're just looking for independent numbers to is
10:27 a factor here . Three is in this tree ,
10:30 it is a factor here , four is in this
10:32 tree , it is also a factor here . Now
10:34 notice that we found that the factor was six ,
10:37 but six is not in this tree , but that's
10:40 okay because the factors are going to be in the
10:42 tree or you can find factors by multiplying different branches
10:47 of the number . So you see how you have
10:49 a three here and you have a two here .
10:52 Well you can find more factors by multiplying three times
10:55 to , to give you six and that's why six
10:57 is a factor four times three is 12 and that's
11:01 a factor here . Um and so basically what you
11:03 do is you build this tree , all of the
11:06 numbers get pulled out of the treat , you write
11:08 them down as your factors and any other numbers that
11:11 pop up when you cross multiply branches of the tree
11:14 . So four times three is 12 , that's in
11:17 there , three times two is six . There's no
11:18 other numbers that you can come up with . All
11:21 right , now let's do the same process for the
11:23 number 15 . Right ? The number 15 . Well
11:28 , the only thing I think of to multiply to
11:29 give me 15 is three times 53 times five is
11:32 15 . And then of course I can go down
11:35 here and I can I can say , well three
11:37 is one times three and five is one times five
11:40 . But you see how these don't get any simpler
11:41 because the only thing that can multiply to give me
11:44 three is one times three . The only time that
11:46 can give me five is one times five and I
11:48 can't do anything more than that because I keep getting
11:51 threes and fives . These numbers here that can't be
11:54 broken apart any further . They're called prime numbers because
11:57 the only thing that can multiply to get three is
12:00 one times three and you can't do anything more than
12:02 that . So we look at this tree and we
12:04 find our factors , we know that one is always
12:06 a factor . We know that the number itself 15
12:09 is always a factor . The only other numbers in
12:11 the tree are three and five , so they're both
12:13 factors and then we can cross multiply the branches three
12:17 times five , but that's 15 and that's already in
12:19 there anyway . So for these problems , the numbers
12:22 were really small . So the trees were not really
12:25 necessary . You can just think through the factors like
12:28 we did before , but when I give you ,
12:31 if I ask you tell me the factors of 98
12:33 2029 then it's gonna be really difficult to figure out
12:37 all the factors , It's actually gonna be way easier
12:39 using a tree like this . So let's go ahead
12:41 and move onto the next problem . The greatest common
12:44 factor here was the number three . Uh and then
12:46 we'll just kind of like , learn this tree method
12:48 here , kind of like on the back burner for
12:51 bigger problems , it will be more useful . Let's
12:54 take a look at the G c f g c
12:57 f Of 10 and 25 , 10 and 25 .
13:03 So , first we want to go through it uh
13:06 kind of without using the tree , let's use factors
13:10 of 10 . We want to figure out what can
13:13 be multiplied together to give us 10 . We know
13:16 that the number one and the number itself . 10
13:18 are always factors so one in 10 are always factors
13:23 . All we have to do is figure out what
13:24 numbers are additional to that . What about two ?
13:27 Well , two times five is 10 , so that's
13:29 factor 33 is not a factor three times three is
13:31 nine . That doesn't work . Four is not a
13:33 factor because four times two is eight . Uh What
13:36 about five ? Well , five times two is 10
13:39 . So five is a factor . What about six
13:42 , 678 or nine ? None of those are factors
13:44 because they can't multiply them by anything to give me
13:47 10 . So the factors are just 1 , 2
13:49 , 5 and 10 . What about the factors of
13:57 25 ? We're gonna do the factors of 25 .
14:00 Next . Yeah , Well , the number one is
14:03 always a factor of every number . And the number
14:05 itself is also always a factor here . So I'll
14:09 put just put away over here , 25 , 25
14:12 is a factor because why one times 25 is 25
14:15 , just like one times 10 is 10 . So
14:17 there are factors alright , is to a factor no
14:21 , three , No four . No , but five
14:24 is a factor because five times five is 25 .
14:26 So I'll put five here . What about 67 eight
14:31 ? Eight times three is 24 99 times three is
14:33 27 10 , 10 times three is 30 . Uh
14:36 10 times to 20 . None of those are factors
14:38 11 , 12 , 13 , 14 , 15 .
14:41 Uh None of those are factors because I can't multiply
14:43 them by anything to give me 25 . So actually
14:46 the number 25 only has three factors 15 And 25
14:50 itself . So the last step is to figure out
14:53 what factors are common . One is a common factor
14:56 five is a common factor , but those are the
14:59 only common factors and the bigger one is a five
15:01 . So we say the g c f is equal
15:05 to five , the greatest common factor of that is
15:08 equal to five . Now we're going to practice the
15:10 tree method to figure out the factors because it will
15:12 be helpful for bigger problems . Let's take a look
15:15 At the # 10 . What times what gives me
15:18 10 ? You can pick whatever you want , other
15:20 than one times 10 . So two times five ,
15:22 that gives me 10 . All right , now ,
15:24 how can I split these further ? Well , actually
15:26 , the tree is pretty much done already because yes
15:29 , one times two is two and one times five
15:32 is five . But you see , I can continue
15:33 getting the same numbers two and five are what we
15:36 call prime numbers , because they can't be broken apart
15:39 anymore . Other than to say , one times two
15:42 is two and one times five is five . So
15:44 that's the tree . So the factors are gonna come
15:47 straight out of the tree . We know the number
15:48 one is a factor , we know the number itself
15:51 , 10 is a factor . The only other numbers
15:53 in the tree are two and five and they're both
15:55 factors and we can multiply branches , but two times
15:58 five is 10 anyway , and that's already in the
16:00 list . So we can get the list , you
16:03 know , as we did right here . Now ,
16:05 let's take a look at the number 25 . What
16:07 times what is 25 ? Five times five is 25
16:11 . But notice that five can't be broken apart anymore
16:14 either . You're just gonna get one times five here
16:16 . So we kind of stopped , we can't do
16:17 any more of the tree . We know that five
16:20 is going to be a factor . One is always
16:22 a factor and the number itself is always a factor
16:25 . So there's only three factors for the No 25
16:28 . So these are very simple trees . But I
16:30 promise you when we get to a larger numbers ,
16:32 it's going to be very , very important to two
16:37 to write the trees down because we'll get all the
16:39 factors that way . So the first two problems ,
16:42 what we call greatest common factor . Now , we
16:45 need to do a very similar problem called least common
16:48 multiple . It's very often confused , but least common
16:52 multiple is different than greatest common factor . So we
16:55 have to talk about that here . What is a
16:57 multiple of something ? A multiple is when you kind
17:00 of skip count by the number . So what are
17:03 the multiples of two ? For instance , The multiples
17:06 of two are when you count by twos . So
17:08 for instance , 2468 10 , 12 , 14 ,
17:12 16 , 18 , 20 forever and ever . You
17:15 can go up as far as you want the multiples
17:18 of to go on forever . What are the multiples
17:20 of five ? 5 , 10 15 2025 30 35
17:24 40 . C What are the multiples of 10 10
17:27 2030 40 50 ? All you're doing is skip counting
17:30 or counting by tens or by fives or by two
17:33 . Those are the multiples . So in our problem
17:36 here , we're going to find something called the least
17:39 common multiple of two and seven . So what we
17:43 have to do is find the multiples of two and
17:46 the multiples of seven . And then we're going to
17:48 find the smallest one that's common to both . So
17:51 it's very similar but different than what we've done before
17:54 . So let's find the multiples of two . The
18:02 multiples of two . All right . What are the
18:07 multiples of two ? We just talked about that .
18:08 Well , we can just count by twos . It's
18:10 too four six , eight , 10 , 12 .
18:16 Now you can just kind of stopped because we need
18:17 to do the multiples of the other one to figure
18:19 out what the answer is going to be . What
18:20 are the multiples ? Yes , of seven . We're
18:26 gonna skip count by seven . Now you don't you
18:28 don't always often skip count by sevens . But you
18:31 know , multiplication is basically skip counting . So think
18:33 of it this way seven times one is seven and
18:36 we count by 77 times two is 14 . seven
18:41 times 3 is 21 . seven times 4 is 28
18:48 . seven times 5 is what ? 35 . So
18:52 this is basically skip counting by sevens , which is
18:55 multiplication seven times 177 times two is 14 . 7
18:58 times three is 21 . 7 times four is 28
19:01 . 7 times 5 . 35 . We don't see
19:02 anything in common yet . But then we realized if
19:04 we keep going in this list 2468 10 , 12
19:07 . The next number is actually 14 and 14 is
19:10 common to both . And notice we're looking for not
19:13 the biggest thing that's common . The least common .
19:16 The smallest thing that's common . So basically this number
19:20 is the smallest number that is common to both lists
19:24 . The smallest number common to both list . So
19:27 the least common multiple Of these two numbers is actually
19:31 14 . The least common multiple of these two numbers
19:34 is 14 . Mhm . And uh you know ,
19:38 honestly at least common multiple is a little easier to
19:40 deal with because you don't have to do any factor
19:42 trees or anything . So we're going to be doing
19:44 a bunch more of these problems to give you the
19:46 hang of it . And we'll also be able to
19:47 speed up a little bit as we go . Let's
19:50 find the least common multiple of four and five .
19:56 The number four and 5 . So let's take a
19:57 look at the multiples of the number four Multiples of
20:02 the # four . We just count by fours .
20:05 Four times one is 44 times two is 84 times
20:09 three is 12 . 4 times four is 16 .
20:13 4 times five is 20 . Now you can keep
20:16 going . But let's just see what we get with
20:18 the other list first . What are the multiples of
20:23 five ? What are the multiples of 5 ? We
20:27 count by fives . That's easy . five , 10
20:30 , 15 , 20 25 and it would be 30
20:35 and 35 , 40 and so on . What is
20:37 common to both of these lists and I look at
20:39 them . The only thing that's common is 20 here
20:41 and 20 here . I could continue the list and
20:44 I would get more numbers that were common , but
20:46 I can already see that the smallest one that's common
20:49 to both list is actually 20 . So the least
20:52 common multiple is actually in this problem . 20 .
20:57 So the greatest common factor , least common multiple .
20:59 Now we're going to continue working more problems and do
21:02 using our trees and things and just kind of getting
21:04 more practice and we'll probably speed up a little bit
21:06 just to kind of get more practice . So now
21:09 we're gonna bounce back to the greatest common factor of
21:15 seven and 28 . So , these are factors remember
21:19 factors we have to figure out what multiplies together to
21:21 give me seven . What multiplies together to give me
21:23 28 ? So let's figure out the factors of seven
21:30 what times what gives me seven ? Well , the
21:32 number one is a factor and the number itself is
21:35 always a factor one time seven and seven . But
21:37 there's actually nothing else that can multiply to give me
21:40 seven . And that's what I was telling you before
21:43 . When the number that you have can only be
21:45 factored by saying the number itself . And the number
21:48 one can be multiplied to give me the number .
21:51 Then we call it a prime number . It can't
21:52 be split apart any more basic than that . So
21:55 seven is what we call a prime number . So
21:57 let's figure out the factors Of 28 . Now this
22:01 one's a little bit tricky because it's a bigger number
22:04 , it's a bigger number , but let's use the
22:06 same process . The factors of 28 . Well ,
22:08 the number one is going to be a factor and
22:11 the number 28 is always going to be a factor
22:14 because the number itself one times 81 times 28 is
22:17 28 . So let's figure out what else can fit
22:20 in here . two is a factor because this is
22:23 an even number , it can divide in there .
22:25 Of course four is a is a factor because four
22:28 times seven is 28 And seven is also a factor
22:32 because seven times 4 is 28 . Now , this
22:35 one might be difficult for you to kind of realize
22:37 . But another factor of this is 14 , 14
22:41 is a factor . You know , when I go
22:42 up from 789 10 , 11 , 12 and 13
22:46 . None of those are factors . They cannot divide
22:48 in here evenly or multiplied by something to give me
22:50 this , But 14 can and you may not remember
22:53 that , but 14 times two is 28 . It's
22:57 not something you might remember . But it's true .
22:59 That's why I'm saying when the numbers get bigger ,
23:01 finding all of the factors might be tough . But
23:04 for now , just trust me that 14 times to
23:06 when you multiply , that gives you 28 and there's
23:09 nothing else bigger than 14 that's going to work .
23:11 So this is the list of factors . What is
23:14 the commonality ? One is common and seven is common
23:18 and the greatest common factor is seven . So we
23:20 say the G c f is equal to seven .
23:23 Now , I definitely want to do the factor tree
23:26 for this . Let's do the factor trees of seven
23:29 . Seven is just one time seven . There's nothing
23:31 else to do . So the factors are one and
23:33 seven comes straight out of there . Now let's take
23:36 a look at the number 28 . Let's figure out
23:38 the factors here . Now , I know from multiplication
23:42 that seven times four is 28 . I know that
23:45 seven can be broken as one times seven . That
23:47 doesn't really go any further than that because seven is
23:49 what we called a prime number . And four can
23:52 be multiplied or broken down into two times two .
23:54 And of course the two can be written as one
23:57 times too , and this too can be written as
23:58 one times too . But I can't go any farther
24:00 than that because I can't split these twos up anymore
24:03 . So let's see if all the factors are here
24:05 . One is a factor Two is a factor four
24:09 is a factor seven is a factor I'm reading right
24:12 out of the chart , 28 is a factor but
24:14 remember I told you you can cross multiply the branches
24:17 seven times two is 14 . So 14 is also
24:21 a factor . If you try to multiply this way
24:24 , seven times four is 28 . That's already in
24:26 the list , two times seven is 14 . That's
24:29 the same number as before . So you see ,
24:30 you can get all the factors from the tree ,
24:32 you read every number out of the tree and it
24:34 goes into your list . And you also try to
24:36 cross multiply branches and that's how we pick up the
24:39 14 . You might have easily missed the 14 if
24:43 you didn't do this . And as the numbers get
24:45 bigger and bigger and bigger and the factors get more
24:47 and more and more , it gets really hard to
24:49 find them all unless you use a tree . So
24:52 the greatest common factor here for this problem was seven
24:54 . Yeah . Alright , next problem . Let's find
24:58 the least common multiple of six and 10 . six
25:03 and 10 . We need to switch back to thinking
25:06 about multiples . Let's find the multiples of six .
25:10 We're going to be counting by sixes . So you
25:12 can think six times 166 times two is 12 .
25:15 6 times three is 18 . You have to know
25:18 your multiplication tables , six times four is 24 6
25:22 times five is 30 and you can keep going .
25:24 But let's for now , let's just see where the
25:26 next list goes , multiples of 10 . Count by
25:29 tens . That's just 10 2030 40 . Go 50
25:35 60 70 and so on . And what is the
25:37 commonality here ? I see 10 here , but I
25:40 don't see 10 here . I see 20 here .
25:42 I don't see any 20 here . 30 . I
25:44 see in both lists . 30 is the smallest number
25:47 common to both . So we say the least common
25:51 multiple is 30 and that's the final answer . All
25:57 right , Okay . For our next problem , we're
26:01 gonna switch back to the greatest common factor of eight
26:05 and 16 . eight and 16 . So let's find
26:10 first . The factors of eight . Alright , What
26:16 can be multiplied to give us eight ? Well ,
26:17 the number one times eight is eight . So the
26:19 number one and eight are always factors . What else
26:23 to is a factor because two times four is 83
26:25 is not a factor , but four can be a
26:28 factor because four times two is eight . So four
26:30 is a factor . What about five ? That's not
26:32 a factor 67 Those are not factor . So ,
26:34 these are all of the four factors of the number
26:36 eight ? What about the factors of the number 16
26:42 ? Of the number 16 ? Well , one and
26:44 16 are always factors the one in the number itself
26:48 ? Because one time 16 , 16 , two is
26:50 a factor because two times eight or 16 , 3
26:54 is not a factor , but four is a factor
26:56 because why ? Four times four is 16 ? Five
27:00 is not a factor . Six is not a factor
27:02 . Seven is not a factor but eight is a
27:05 factor . Why ? Because eight times two is 16
27:07 . And if you go up from there 9 ,
27:09 10 , 11 , 12 , 13 , 14 ,
27:11 15 , none of those are factors . They cannot
27:12 multiply to give me 16 . What is the greatest
27:16 thing ? Common one is common to is common .
27:19 four is common and eight is also common and this
27:23 is the largest of the common one . So the
27:25 greatest common factor is equal to eight . All right
27:31 now , just to get practice , we're gonna use
27:33 these factor trees . What about a tree for eight
27:35 ? We know that two times four is 84 can
27:39 be broken down further into two times two . I
27:41 could break down the tubes but it's not gonna add
27:44 anything . So I look at the tree and I
27:47 say to is a factor four is a factor one
27:51 . And the number eight are always a factor and
27:53 that can cross multiply the branches two times four is
27:56 eight , that's already in there . So there's nothing
27:58 else that the tree gives me . Let's take a
28:00 look at 16 . All right now , you can
28:03 do whatever you want for 16 . You can do
28:05 four times four , but let's just do two times
28:07 eight . I mean you can pick whatever you like
28:09 . Eight can be further broken down into two times
28:12 four and four can be broken down into two times
28:16 two . Now you can choose different like four times
28:18 four and so on . But you're going to get
28:20 the same place for 16 . The number one and
28:23 the number 16 are always factors . two is a
28:26 factor Four is in the tree . That's a factor
28:29 eight is in the tree . That's also a factor
28:32 . And you try to cross multiply two times four
28:35 is eight , that's already in there and there's nothing
28:37 else to cross multiply . So these are all of
28:39 the factors of 16 . And so the common .
28:42 The greatest common is the number eight . Alright ,
28:46 the next problem . Let's take a look at the
28:48 least common multiple of uh Sorry , not eight .
28:51 I don't know why I have eight in my mind
28:53 . The least common multiple of four and 10 .
28:56 So let's find the multiples of four . First .
29:00 All we need to do is skip count by four
29:03 . So four times one is four , four times
29:06 2 is eight , four times 3 is 12 .
29:09 Four times four is 16 . four times 5 is
29:12 20 counting by force . Next we have to say
29:17 , what are the multiples of 10 ? Now ?
29:20 These are actually easier . Everybody knows how to count
29:22 by tens , 10 , 20 30 40 and so
29:27 on . And you can see right away that we're
29:30 trying to find common things and we're trying to find
29:32 the least common . Now , we're trying to find
29:35 the things common . And the only thing I really
29:36 see is a 20 and of course that's the smallest
29:39 thing that's common . None of these other numbers are
29:41 common to both lists . So 20 is the least
29:44 common multiple ? These common multiple is 20 . Alright
29:50 , next problem . Only two more problems . Actually
29:54 , let's take a look at it over here .
29:55 Let's find the greatest common factor of 12 and 32
30:04 , 12 and 32 . So , what I think
30:07 I want to do here is let's find the factors
30:11 of 12 and let's find the factors mhm . Of
30:16 32 . But you see how 32 is kind of
30:18 a large number . I think I want to use
30:19 the factor trees straight away for this problem . Let's
30:21 do it for 12 . So 12 . What times
30:25 what is 12 ? You can do two times six
30:27 , but let's just do three times four , mm
30:30 Four can be broken down into two times too .
30:33 But you see , I can't break three anymore .
30:35 It'll be just one times three and two . I
30:38 can't break anymore because it's just gonna be one times
30:40 two , so I can write them if I want
30:42 . But I don't need to I know that the
30:44 number one is a factor and I know that the
30:46 number 12 is the number itself is always a factor
30:49 one times 12 is 12 . Now , what else
30:52 comes out of this tree ? The number two is
30:54 a factor comes straight out of the tree . The
30:55 number three is a factor comes right out of the
30:58 tree . The number four is a factor comes right
31:00 out of that tree . Now , I cross multiply
31:03 branches three times two is six . That's another factor
31:07 . So , what I'm gonna do is scoot this
31:08 12 over here . six is also a factor .
31:13 All right . And also you can try three times
31:16 to a 6 . 4 times three is 12 ,
31:18 that's already there . So , we read all the
31:20 numbers out , put them in and cross multiply branches
31:23 . These are all of the factors . Make sure
31:25 you agree . Two times six is 12 , 3
31:27 times four is 12 . 4 times three is 12
31:30 . 6 times two is 12 and 12 times one
31:32 is 12 . So , these are all of the
31:33 factors . Now , let's take a look at the
31:36 factors of the number 32 . So this is a
31:39 larger number , but you can pick whatever you like
31:42 , but let's just do four times eight is 32
31:46 . Now , the four can be broken down into
31:47 two times to the eight , you can break it
31:50 down into two times four and before you can break
31:54 this out to two times to you see how I
31:56 get it all down to twos , because I two's
31:59 are prime numbers , I can only break it into
32:01 one times two . So I'm just gonna stop here
32:03 . Now , I know that the number one is
32:05 a factor and I also know that the number 32
32:09 is also a factor . The number itself , they're
32:10 always factors . What else do I have in this
32:13 tree ? The number two it says is a factor
32:15 . The number four it says is a factor .
32:19 What else do we say ? It says the #
32:21 eight is a factor . I'm going to read that
32:24 out . So I have 24 and eight is also
32:27 a factor . I've written all the numbers out of
32:28 the tree . Now let's try to cross multiply eight
32:31 times two is 16 . It says that that's another
32:35 factor . Is that ? All of them ? four
32:37 times 4 is 16 , same number four times two
32:39 is eight , that's already in the tree , Eight
32:42 times two is 16 . Yeah , there's no other
32:43 numbers here . So the factors are 1248 16 and
32:49 32 . The 16 here , you might have missed
32:52 if you didn't use the factor tree , if you
32:53 were just thinking through , you might might have gotten
32:55 it . You might not have . I mean depends
32:57 on how you're feeling that day , but write the
33:00 numbers down out of the tree and then also cross
33:02 multiply any branches . So what are the factors in
33:05 common ? one is common to ? Is common for
33:09 ? Is common . What else is common ? Nothing
33:11 else is common above that . So four is the
33:14 greatest common factor . Greatest common factor . Four .
33:18 Greatest common factor is the number four . All right
33:24 . Here is our very last problem . Let's take
33:27 a look at the least common multiple of four and
33:30 6 . So multiples Of four is what count by
33:36 force . four times 1 is four . Four times
33:39 two is 84 times three is 12 . 4 times
33:43 four is 16 . Just stop there multiples of the
33:47 number six . I can write the number six correctly
33:50 . Sorry about that . The number six , six
33:53 times one of 66 times two is 12 . 6
33:55 times three is 18 . And you can keep going
33:57 if you like . What are the commonalities ? The
34:01 only thing I see here is the 12 and of
34:02 course that's the smallest one that's common to both list
34:05 . If I keep going , I'll get more numbers
34:07 common , but 12 will be the smallest least common
34:10 multiple is 12 . So I hope by doing all
34:15 of these problems , you can see that even though
34:17 we have to think about the least common multiple and
34:20 the greatest common factor , even though we have to
34:21 work for it , the process is the same for
34:23 every one of these problems . A lot of people
34:25 get them confused and mix them together . And I'm
34:28 really just trying to help you get over that by
34:30 solving a lot of problems I'd like you to go
34:32 back through and solve every one of these yourself and
34:35 I really do want you to write the factor trees
34:36 down because even though you might not need it for
34:39 the small numbers , I guarantee you eventually you'll get
34:42 a problem that you cannot do without a factor tree
34:45 because then the numbers are so big , you won't
34:48 be able to find them all . You'll easily miss
34:50 the miss a factor . So I'm trying to get
34:52 you in the habit . Plus they're kind of fun
34:53 to do , you know , like a little drawing
34:55 . So I would like you to do those Then
34:57 follow me on part two where we will continue skills
35:00 with factors and multiples .
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