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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