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so hello there and welcome to the
tutorial my name is Tanner Baki and this
time we're going to be going over
another general education video so
actually this was a question sent to me
by one of my viewers and here's a clip
of their
question hi Tammy I'm Madison and I have
a little math question for you what is
LCM and GCF how can I find an LCM and
GCF of two numbers also is there any
relationship between the numbers of LCM
and GCF
I will be looking forward to the next
video explaining what LCM and GCF is
bye so now that you've seen the question
I'm going to try my best to answer it I
hope it answers a question Madison let's
get started uh so today we're going to
be talking about the LCM and GCF uh and
before we go any further what does LCM
and GCF actually stand for in the first
place well LCM as you can see stands for
lowest common
multiple which I'll explain in a
minute and GCF stands for
greatest
common factor so essentially the
opposite completely of the
LCM now we can ignore this part for now
and let's look at this these are two key
words that will really help
us when
finding uh when actually trying to do
the LCM of the
GCF so there's two things here the
multiple and the
factor and so you can have a multiple or
a factor of a number for example the
number
eight I'm going to list out its
multiples and its factors and then show
you exact what uh the factors and
multiples mean so for eight you could
have something like uh 1
2 4
and8 for its multiples you could have oh
I don't know
eight 16
24 32 dot dot dot they can keep going on
forever okay now let me explain exactly
what these things are why have I listed
these numbers exactly what do these have
to do with the number eight well
factors are numbers that 8 can be
divided by for example you can do 8
divided 4 and it would give you two 8
divided by two and it would give you
four etc etc for all its factors and the
multiples are the opposite they are the
numbers that can be divided by eight so
for example let's just choose a random
one uh this one 24 okay so I landed on
24 24 ID 8 three perfect so it can
easily be divided and so can
this so just to sum it all up a factor
uh is a number that eight can be uh
divided by and then the multiple in this
case uh is the number uh that can be
divided by eight you could replace this
with any number and then its multiples
and factors would
change and then stuff like that now you
must have seen a pattern here the
factors can only
be less than or equal
to 8 but the multiples can only be
greater than or equal to
8 so uh this is basically how factors
and the multiples will work and so if
you didn't notice already I put a dot
dot dot here what this means is that the
multiples will go on Forever Until
Infinity however our factors will stop
right here at
8 so that's how that
works now that we've gotten that out of
the way uh let's actually start using
the LCM and the GCF of multiple or more
numbers and just before I go on let me
just clarify that an a word for this
sort of a set would mean this is a
finite set meaning there's a LUN limited
number in this however here there's
infinite the in basically uh meaning the
opposite of finite meaning this is
infinite it'll keep going on and on and
on and on and on and on you could do 8 *
5 million and you would still get an
answer you could do eight times Google
Plexi and then you'd still get an answer
which is a real number by the way okay
anyway getting back to the point let's
actually see how you find gcfs and
lcms our main way here is of course
going to be something called
factorization
trees and before I actually explain
factorization trees to let me explain
what lowest common multiple and and
greatest common factor actually
mean so for example here uh let's just
say I wanted to find the LCM uh of oh I
don't know uh 4 and
8 now what I do is i' list out the
multiples for four I'd list out the
multi for eight so for four there could
be um there could be four there could be
eight there could be 12 there could be
16 etc etc that'll keep going on till
Infinity for eight there could be 8
16 uh there could be 24 there could be
32 etc etc again onto
Infinity now as you can
see first of all we have the word
lowest in
LCM and we also have have
common so you must have found out that
we need to find the lowest common factor
I mean a multiple quite literally from
both of these sets so eight it's in this
set so our lowest common multiple will
be
eight and just as a little rule quickly
the
LCM can only be greater than or equal to
the bigger number in the Set uh so in
this case it's
eight if it were for example uh 8 and 20
it would be
20 now let's try to find the
GCF so for example here I'm going to uh
write down all the
factors for four
and8 oh just to clarify uh we found the
LCM
here to be 8 now let's find the
GCF so let's uh list down the factors
for four and eight so for the factors
for four would be
one two and that's really and also
four and for eight we could have one two
four and also eight so as you can see
the greatest factor that's common in
both couldn't be eight because that
doesn't come in uh in four but it is
four because it also comes in four so as
you can see in this case our GCF
sorry is going to be
four so now you must have seen another
pattern our GCF can only be less than or
equal to four in this
case so that's how that works anyway
apart from this example now you actually
know what an LCM and what a GCF is now
you might be wondering what's a really
efficient way to do this without having
to write all these factors and all these
multiples and everything this is very
manual method it doesn't work with
larger numbers and it's just overall a
very tedious
process so how can I
make a way to find the same GCF and
LCM easily so yeah we can use prime
factorization now let me give you a few
examples of that so let's start with uh
finding the G uh the LCM of a few
numbers such as uh oh I don't know four
and 10 so for example four and 10 now
again we really do not want to list out
all the multiples all the factors find
the answer it's just really repetitive
sometimes you miss one and it just keeps
going on and it's very unsuitable for
larger numbers like
5,659 with
3,291 that just wouldn't work with
listing out all the multiples and
factors so what we do in this case is we
do something called prime
factorization so what we do here uh is
you must know what a prime number is
it's a number that can only be divided
by one in
itself so I'm going to give you an
example of something called prime
factorization so as you can see have
first of all let's look at this
number what * what is equal to this
number 2 * 2 is equal to this number so
I'm going to write down two and two in a
tree
format now basically there's no more
that we could continue because two is a
prime number two is a prime number we
couldn't continue our tree however in
10's case you also cannot continue the
tree
it stops at two and five because 2 * 5
is 10 and so since these are both Prime
we can't continue to create more numbers
in a
tree just so that you aren't uh confused
let me just show you uh the tree of some
something like eight the reason I'm
going to show this to you is so that you
really get the concept of a tree in this
sense so what's uh 2 * 4 8 so I'm going
to write down 2 two and
four now two is prime but four is not
this uh is
uh forgot what that's called anyway uh
this is prime uh this is not prime sorry
uh so then we can continue two and
two uh so yeah now we've factored it
down to only literally uh prime numbers
so now that it's only Prime we can't
continue and we have successfully
created a prime
tree uh so now uh since we can't do any
of this with 10 and four we can only
have this oh yeah just remember the name
this is a composite number it's not
prime it's composite okay remembered
okay now let's go back to this since uh
there's no more down we just stay at one
level here in the tree uh and that
really works
out anyway let's take a look at this now
since since we have treed down our uh
two
numbers what we can do now is in order
to find the LCM let's take the number 10
and let's just say
2 * 5 because that's what it's what
that's what's written down here then
under that I'm going to write 2 *
2 but when you do the this sort of a
thing there's a little limitation I
guess you could call it
and so what
happens is okay look at this you have to
take
erase if two of these numbers are not
the
same uh and what you have to do is put a
little square
here another times and then the other
number because there should never be two
numbers that are not the same uh on top
of each other
uh so yeah that's how that works uh and
so just so you see it a little bit more
clearly I'm going to say 10 is equal to
uh for the sake of
it okay and then I'm just going to do
this and as you can see it's that easy
to be able to uh create something like
that anyway now as you can see uh we
have a little multiplication statement
but how do we get our LCM from it all we
really need to do is say okay two and
two they're the same so write down a
two then a multiplication
symbol five and blank so we're going to
write down
five blank and two or we could also
write down blank so blank and two we are
down to
two so our
LCM is equal to 2 * 5 * 2 so let 2 * 5
10 * 2
20 so our LCM has has successfully been
found to be
20 so that's our LCM now how do we get
our
GCF so our
GCF will just be equal to two how you
may
ask that's
because first of all and just that you
can uh I can draw some arrows for you
I'm going to write it above so for
example here GCF
now as you can see two and two they are
common so I'm going to write two then a
times five and blank not common blank
and two not common so there's no more
numbers at the end we remove our
multiplication and as you see our GCF is
going to have to be
two so we have successfully found that
the GCF for 4 and 10 is 2 and the LCM
for 4 and 10 is 20
anyway continuing on with this uh Rule
now I'm going to show you a few more
examples of prime factorization finding
the LCM and GCF with that
method so now just take a few numbers
for example 6 and
10 okay now we have 6 and 10 let's prime
factorize
them that's done pretty
easy pretty
easy now let's find the
LCM so I'm going to note down two and
five for number
10 and for six we note
down two and
three however again there can ever be
two different numbers under each
other so what we do
is we
erase we make a little
box make a little
box and then put the actual number which
in this case will be
three so now that we have both of our
numbers written
down over here all we can do is say okay
two and two that's a
two times five in blank that's a five
time blank and three which will be a
three so our
LCM will be equal to this so now what's
2 * 5 that's 10 * 3 that would be 30 so
30 is our
LCM now we found our LCM let's find our
GCF again there's only one number that
is common throughout all of these so we
can assern that our GCF is that one
common number which in this case is
two so now what we need to do uh is do
some more examples so let's just do
something a little bit more complicated
now uh 10's not that interesting because
um sorry about that uh they're just one
down in a tree and so that really isn't
very fun if you think about it and you
really wouldn't get those many questions
about that
either so now let's do something like 5
and 12 for
example so maybe 5 12 now five is
already Prime so I mean we could write
down one and five uh but there's nothing
really that we would need to do about
that anyway but 12 that's interesting
because to get 12 you would multiply two
and six and to get six you would get two
and three so as you can see we have a
variety of different numbers now so now
what I'm going to do is I'm going to
come over here I'm going to say for
12 uh we have two uh so as you can see 2
* 2 * 3 because those are all the prime
numbers that we got 2 * 2 * 3 will give
us 12 because 2 * 2 that's 4 * 3
12 then we take our
five right now the thing is we can't
just write down a five here so what I'm
going to do is I'm GNA put a blank and
as you can because there's a two in
front of it and that's not a five so
they can't be the different they can't
be different so we can't put that there
so we have to do blank times another
blank times another blank times a
five and that is our valid
multiplication
statement so then I'm just going to do
this and so now as you can see I'm going
to just find the LC M by doing
LCM is equal to
2
* two times as you can see because two
blank two two blank two three blank
three and
then blank five five so we say 2 * 2
that's 4 * 3 uh 12 uh * 5 would be uh
24 wait no uh that would
be forgot what it was uh let me
calculate this 60 sorry got confused
there anyway uh but 12 * 5 would be 60
and so we can definitely assern that our
LCM is
60 now let's find our GCF but wait a
minute no matter what number you look at
here there's either a box box below it
or box above it this means that there's
no common
number so when there's no common number
the only thing that you can think is
that the GCF will have to be one cannot
be any other number due to the fact that
none of these have a common number in
them
anyway now that that's done uh let's do
some more examples uh let's just uh
finish it off with two more examples
this time we're going to do 8 and 12 so
12 is already there so then we don't
need to erase that though we can erase
this and
stuff yeah so I'm going to give you 8
and 12 now which I'm sure you will all
love this example because it's just a
perfect mix between uh hard and easy at
the same
time so then I'm just going to do this
and then stas this because I mean we're
still using 12 we're just putting in an
eight
so that'll work now let's prime
factorize 8 sorry about
that anyway let's prime factorize eight
so we take our eight what's 2 * 4 eight
so 2 and
four what's 2 * 2 4 so we have
successfully Prime factorized 8 by doing
2 * 2 * 2 is equal to 8 and that's a
completely valid multiplication
statement so now what I'm going to to do
I'm going to come over here and going to
say 8 is equal to uh 2 * 2 * blank *
2 because that's actually what we're
doing with 8 and 12 how to find our LCM
however I want to say
LCM is equal to 2 * 2 * 3 * 2 so 2 * 2
that would be 4 * 3 that would be 12 * 2
that would be 24 so our answer here is
24 for our
LCM that was a pretty nice uh way to
find our LCM now let's find our
GCF so as you can see there's two common
numbers in this uh so there's two and
two so we do GCF I'm just going to write
it here if you don't mind so
GCF is equal to 2 * 2 because there
those are the only two common
uh numbers there so the GCF is going to
be equal to 2 * 2 which is equal to four
so our GCF is successfully
for now if you watch the question
carefully uh yes uh Madison did ask me
uh the how to find the GCF and LCM but
she also asked me if there was a
relationship between the numbers and
their GCF and LCM and yes there is so
let me explain that to you right now
so now I'm going to be explaining the
relationship between the two numbers and
their LCM and their GCF order to do this
since I don't want to bore you and spend
the next 10 hours writing like doing
tons of questions I'm just going to
write down lots of gcfs and lcms along
with their numbers so this is going to
be number one number two their
LCM and their GCF
now what I'm going to do uh is I'm just
going to note down lots of number one
number twos and calculate older lcms and
gcfs and let's see if we can find a
pattern let's do something like uh 10
that would be a pretty good
example and so I'm just going to
calculate its
LCM which would be 40 so the LCM would
be 40 here and the GCF would be uh two
perfect so our LCM and GCF is 40 and two
I'm going to continue this and write
down lots of
numbers still be
40
be okay
four perfect uh then we can do something
like uh three and
four the LCM would obviously be 12
uh and their GCF would be one and I'll
tell you how I calculate this calculated
this in just a minute let's do something
like seven and
eight
do
56 and
one you'll see how I calculated these so
fast in just a minute
now 6 and
10
uh 30 and
two okay uh and then we could do uh two
more examples I guess or actually a few
more examples quite a few uh 4 and
20 don't think I have enough space for
the other pattern that I was going to do
so as I know 4 and 20 would obviously be
20 because uh that's just how this works
and again I'm going to be telling you
how I calculate these so fast in just a
minute and then this would be
four then if we were to have something
like 6 and 24
this would be 24 and this would be
six I think you might just start seeing
a pattern now but you might just want to
wait for a while and see what I really
come up with
next so now that we're done 6 24 24 and
6 uh let's go to another example uh
seven and 11 I guess 7 and
11 actually no 7 and 11 would be too
complicated for now or what it yeah I'm
just going to give you a different
pattern which is five and six would give
us 30 and
one so then after five and six we have
something like five and
seven then our answer would definitely
be
35 and
one so then I will do a few more later
but that is if I could possibly fit them
uh so there are a few more patterns that
I should should be explaining but those
will be explained in just a
minute so now as you know we have number
one number two our LCM and our GCF now
what's the relationship between these
numbers you may
ask well let's just look at this what's
40 * 2 we all know that's 80 because 4 *
2 8 i 0
8 what's 8 * 10
80 so if 8 * 10 is 80 and 40 * 2 is 80
we can say that number one * number two
is equal to LCM * GCF so this is our
final rule uh number one * number two is
equal to LCM *
GCF that's how that works that's a
really interesting concept and it'll
really help us uh in finding the LCM and
GCF and and also find number one and
number two now let me just show you that
this exact same pattern works with all
of these numbers and then I will
actually show you a few patterns in
between all these numbers which you
might not have
realized okay so let's take something
like uh number one and number
two so 8 and 20 what's 8 * 20 160 so 8 *
20 is 160 40 *
4 160
so these two are equal to each
other 3 * 4 obviously 12 12 * 1 very
obviously 12 then we can assern that
they're
equal 7 * 8 56 56 * 1 56 and so we can
say that these are equal this will keep
going on for example 6 * 10 uh 60 30 * 2
60 these are equal this will always work
100 % of the
time no matter what 4 * 20 though just
to clarify if you have any possible
questions 80 okay this is 80 20 * 4
80 and I'm just going to say that uh
these are equal because 6 * 24 uh which
I'm not going to do right now because
that would be way too much to calculate
uh but I mean if you were to do this it
would be equal to this due to our rule
saying so so we don't even need to
calculate it we could keep going on 5 *
6 30 30 * 13 so then these will be
equal 5 * 7 35 = 35 *
1 that's how that works and as you can
see this rule will actually help us even
though you might not think about it
because let's say you do have your LCM
but you want to still calculate your
GCF what what would you
do all you need to do is to find the
GCF you find number one times number two
divide or actually
divided sorry about that a little bit
too excited I guess
anyway okay so number one time number
two divided by the
LCM is going to be equal to our
GCF so so for
example let's find the GCF let's pretend
we didn't have this
answer okay let's pretend we didn't have
the
GCF we only have the LCM and our two
numbers so our
GCF is equal
to 8 *
10 divided by
40 so so what's 80 / 40 obviously our
GCF will be equal to 2 so our GCF is
equal to
2 so that's how that works and I'm just
going to give you a few more examples
then I'm going to explain uh two or
three different patterns that I noticed
uh when finding your GCF and your
LCM okay so let's just choose one at
random okay so this
one let's just try finding the LCM this
time okay so we don't have our LCM we
have our GCF though so what we're going
to do is we're going to do
LCM is equal
to 6 *
10 /
2 so now we do 6 * 10 which would be 60
/ 2 30 so our LCM will obviously be 30
and so we can fill in our 30 over
there that's how you would find the LCM
and GCF using the other number and this
actually also works if you want to find
one of the other numbers so for example
over here we didn't want to uh okay
let's just say we didn't have
10 okay we don't have
10 but we have 30 two which are our LCM
and GCF and we also have number
one so what we're going to
do is number
two uh number
two is equal
to 30 *
2 ID
6 so 30 * 2 what would that be that
would be 60 then divide that by six and
you get 10 so our number two will have
to be 10 which is correct so it will
work even for number one number two LCM
or GCF as long as you have one of the
other or vice
versa okay now that that's done let me
explain three patterns that I
have first of all you may have already
noticed this if the numbers that you're
calculating for are one
apart their LCM is equal to them to
multip applied and their GCF is 1 for
example 3 and 4 what's 3 * 4 12 their
LCM is 12 and their GCF is 1 7 * 8
what's 7 * 8 56 their LCM is 56 and
their GCF is 1 and again this is a rule
that will work 100% of the time no
matter what as long as the two numbers
are one
apart so I can give another example like
five and six over
here uh 5 * 6 30 and so our LCM is 30
and our GCF is
1 so that's basically how that works uh
and so that's one of the patterns now
let's see another pattern and you might
have really not seen this because I
haven't shown you an example of this yet
however now I'm going to erase a few of
these uh till there I
guess and so this pattern is really
interesting
so you're really going to like this
one okay watch this if I have number one
as like seven and 11 like the earlier uh
one that I was giving you it's LCM will
be 77 and its GCF will be
one but you wait you must be thinking H
these aren't one
apart like what's happening well I'm
going to give you a few more examples
maybe you'll find out for three and
seven Guess 35 you're
right all right
sorry I made a
mistake I I made a mistake
anyway forgot to do that and it's
actually 21 but it's GCF will always be
one and I can guarantee to you that the
next one will also be one and you'll see
how I thought that in just a minute and
I I don't even know these two numbers
yet and I know that they're it's going
to be one so I'm just going to think and
so five and
seven okay so it's LCM would be
35 now I want you to guess what the
pattern
is the pattern is that if the number if
number one and number two is
prime then the LCM will always be their
multiplication and their GCM will be one
that's how simple that is same thing for
this because three and seven are
prime uh 3 * 7 21 21 and 1 for the LCM
and
GCF 5 and 7even Prime 5 and 5 * 7 35 35
for the LCM one for the
GCF that's how that actually
works so those are the patterns that uh
we could assern from these uh little uh
numbers and LC M and I really hope that
answers your question
Madison cuz uh that was pretty much I
guess uh it for this video uh and wait
just one more thing for I forgot to show
one last pattern to you let's just uh do
this number one number
two LCM
GCM so the last pattern is let's just
say we
have 4 and
20 and also 6 and
24 I can immediately guess
these if you can't guess why I don't
blame you it's a pretty hard pattern to
find you do realize that it's basically
just a mirrored reflection of the number
one and number two but you don't
understand why yet let me explain
if the bigger
number is a multiple of the smaller
number then the LCM will be equal to the
bigger number and the GCF will be equal
to the smaller number that's just how it
works so for example 4 and 20 20 is a
multiple of
four so then our LCM will be the bigger
number which is 20 and our GCF will be
the smaller number which is four and
that's how that rule works that is the
last really last rule so I guess that's
pretty much it for this video uh and
yeah uh now I'm going to say my outro as
always however if you sorry about that
that actually hurt
anyway anyway there is a bit of a longer
example at the end of this video so if
you want to stick around please do uh
but as always for the people that don't
want to stay around I'm just going to sa
my outo so yeah subscribe to my channel
if you're new like the video if you
liked it comment down if you have any
questions or any suggestions for any
apps uh and that's going to be it uh
goodbye to everyone who doesn't watch
the doesn't want to watch the longer
version uh but hello to everyone who's
joining in for the longer version so uh
let's try our best to find the LCM and
GCF for two numbers pretty huge numbers
like uh 24 and
64 okay now how would we go about doing
this we're going to do is we're going to
use the same method as always uh which
is basically just Factor these down use
multiplication it's going to be a little
hard uh to do because there's so many uh
factors but again this is not a really
uh what you call it a realistic example
you wouldn't ever have to do this you
would usually use a calculator or
something uh but just in case who ever
posed with this question uh it probably
won't even fit on a few pieces of paper
but I'll show it to you anyway so for 24
what we do is uh 2 * 12 is
24 uh then 2 * 6 is
12 and then 2 * 3 is 6 so we have
successfully Prime factorized 24 for 64
however it's a bit different 2 32
because 2 * 32 is equal to
64 then 2 and 16 uh because that's what
that's equal to uh 2 * 16 is 32
then uh to get 16 we do two and
eight then to get eight we do two and
four and to get four we do two and
two now what we need to do is we need to
write down our multiplication uh
question uh and then just solve it how
simple it is so what I'm going to do is
I'm going to come over here 1 2 3 4 5
six there's six twos in 64 so I'm going
to say 64 is equal to 2 * 2 * 2 * 2 * 2
* 2 1 2 3 4 5 6 okay so there's six uh
twos in
64 anyway now 24 I say
24 1 2 3 two so I'm going to say 1 2 3
then times a box times a box Time a box
Time a three
so now we have to find our LCM and we're
going to do that by multiplication as
always so then LCM is going to be equal
to 2 * 2 * 2 * 2 * 2 * 2 *
3 now again you could just go 2 * 2 is 4
then 8 then 16 then 32 then 64 then 192
but that's boring and plus if you didn't
already already know the answer to this
question you wouldn't want to go through
this entire thing and also if this were
much bigger for example like 10 more
twos you wouldn't keep on doing that so
I broke it down into multiple little
sections which I like to call a
multiplication tree so then we just take
three parts of this question 2 * 2 2 * 2
2 * 2 so the answers to these will be
four
now what we do is we find the answer to
these two what's 4 * 4 16
obviously okay then we bring the four
down uh down
here we have to find 16 * 4 so I bring
that down to
16 * 2 * 2 same thing it's easier to
solve too so 16 * 2 32 * 2 64 so then
our answer here is 64
four then finally what we do we bring
down our three over here so 64 * 3 which
sometimes You' need a calculator for
unless you're really good at
multiplication mentally uh but I know
that the answer here would be
192 so our LCM in this case for 24 and
64 would
be92 now if you wanted to find the GCF
however what you would do
is again you would take the common ones
and just to do this easier I'm going to
make another box so there's three twos
that are common so I'm going to say the
GCF is equal to 2 * 2 * 2 there's only
three numbers common so 2 * 2 4 * 2 8 so
our GCF is equal to
8 and our LCM is 192 for 24 and
64 so that's how it's solve
that and yeah that was pretty much it
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