Hello and welcome to yet another video on Quantitative
Aptitudes.
In today's video, we will learn how to calculate LCM and
HCF of more than one number.
As we did in the last video, we learned how to calculate
factors and multiples of any particular number.
Like we learned factors of 4, multiples of 2 and multiples
of 3.
In the same way, today we will discuss the following topics
which will be helpful to you.
That is, how to calculate LCM and HCF.
The full form of HCF will be Highest Common Factor.
Here, the word factor is used.
We saw the definition of factors yesterday.
Similarly, what will be the LCM?
LCM will be Least Common Multiple.
So, when will we find the Least Common or Highest Common?
When we have more than one number, we will find the Highest
Common Factor.
And what is the Least Common Multiple of these two or more
than two numbers? So, see their full form.
Highest Common Factor of HCF.
And we call it by another name.
That is GCT.
GCT is also another name of HCF.
We will have the full form of GCT.
Greatest Common Divisor.
We will have the full form of LCM.
Least Common Multiple.
So, now we will find out how to calculate these in
questions.
So, let's take a very small example.
First, see again that Highest Common Factor.
Why are we finding out the Highest Common in factors?
Yesterday, we studied in factors that how many numbers of
factors are there in any number?
Are these numbers finite?
That means we can count them.
There must be some limited number of factors of some number.
So, we can find out the highest in that.
Because if we have a set of numbers, a finite set, then
which will be the biggest number in that?
Which will be the smallest number in that?
We can tell that obviously.
That is why we can tell the highest factor in factors.
Lowest will be 1 for all.
Because 1 is the factor of all.
So, it will not make sense to ask about lowest.
That is why Highest Common Factor is asked.
Similarly, what I told you in multiple?
That multiples are infinite in numbers.
The multiples of any number keep on going.
There is no end to them.
So, will we be able to find out the highest multiple in
that?
We will never be able to find out.
Because it is never ending.
That is why in case of multiple, we find out the least
common multiple.
That means we find out the smallest multiple of any two
numbers.
So, let us learn this from questions.
How will we find out in questions?
If someone is giving us 2 or 3 numbers,
How will we find out their LCM and HCF?
Suppose you have numbers.
One number is 5.
Second number is 15.
Third number is 6.
What do we have to find out?
We have to find out their LCM.
So, how will we find out their LCM?
First of all, prime factorize everyone.
5 is already a prime number.
So, we will write 5 as well.
What can we write 15 as?
We can write 15 as 3 x 5.
3 is also a prime number and 5 is also a prime number.
How can we write 6 as?
Similarly, 2 x 3.
2 is also a prime number and 3 is also a prime number.
Now, look at this.
We are finding out their multiple.
Now, we are finding out their LCM.
Before that, let me teach you HCF.
We have to find out their LCM and HCF.
First, let us learn HCF.
We have prime factorized this.
What is HCF?
Highest Common Factors.
So, what are these factors?
Obviously, 1 is also a factor.
So, we will not look at these factors.
We will look at the numbers other than 1 and 1.
We will look at prime numbers.
Now, look at this.
What is common in these numbers?
Is there anything common in these numbers?
This has 5.
So, 5 and 5 are common in both these numbers.
But, is 5 coming in 6? 5 is not coming in the factor of 6.
So, 5 is not common in these numbers.
Similarly, look at 3.
3 is also coming in the factor of 15.
But, it is not coming in 5.
But, it is coming in 6.
So, we got 3 in both these numbers.
3 is common in 15 and 6.
But, we did not get 3 common in 5.
So, 3 is also not possible.
And, 2 is only in 1 and 6.
So, it is not common in any of these numbers.
So, it is not common in other numbers.
So, what is common in these numbers?
We have to look at the common factor.
And, the biggest common factor in it is 1.
Because, 1 is the common factor of all numbers.
So, 1 is the common factor of all numbers.
So, the common factor of all numbers is 1.
So, 1 is the highest factor in all numbers.
And, 1 is the lowest factor in all numbers.
Because, apart from 1, there is no other common number.
So, what is the HCF of all these numbers?
What is the HCF of all these numbers?
HCF of 5, 15 and 6.
Now, this was the question in which you did not get any
common factor.
So, when we did not get the common factor, then the common
1 is the common factor of all numbers.
So, we got 1.
So, yesterday we studied one thing.
Co-prime numbers.
When were the co-prime numbers?
When we have only one common factor in them.
That is 1.
So, indirectly you got this example of co-prime numbers or
not?
That what are these three?
They are co-prime with each other.
That means, they do not have any other common factor except
1.
So, what happened to these three?
They became co-prime.
Now, we did this example in which we are not getting any
common factor.
I will tell you one example.
Suppose, we get a common factor.
What should we do?
Suppose, we have numbers 6, 18 and 24.
Find HCF.
What we have to find out?
We have to find out HCF.
So, first of all, what should I do?
Prime factorize.
We just saw 6.
2 into 3.
How can we write 18?
We can write 18 as 2 into 3 into 3.
2 into 9 is 18.
So, we have split 9 into 3.
So, 2 into 3 into 3.
How can we write 24? 24 is 2 into 12.
So, it becomes 2.
Then, split 12.
What will be 12? 2 into 6.
So, we get 2.
Then, split 6.
That is 2 into 3.
So, what is the prime factorization of 24? 2 into 2 into 2
into 3. So, power of 2 is 3 into 3.
This is done.
Now, see what is common in this? First of all, 2 is in 6.
2 is also in 18.
And 2 is also in 24.
So, 1 is common.
So, write it once.
Whatever common factor is coming, write it once.
If 2 is common in 3, then write 2.
Then, 3 is common in 6.
3 is also coming in 18.
And see, 3 is also coming in 24.
So, 3 is also a common factor of these 3.
So, write it as 3 into 3.
Now, what is remaining common?
How will the remaining common come?
Because 6's factors are over.
The remaining factors of 18 are 3.
The remaining 2 and 2 of 24 are 3.
But, we cannot take them as common.
Because, they should have come in 6.
Only then, we could have taken them as common.
So, what are the common factors? 2 and 3.
So, their multiple will be 6.
So, what will be the highest common factor of these 3?
Hcf.
Their highest common factor.
The highest common factor of these 3 is 6.
See, 6 is coming in 6's table.
So, it is a factor.
18 is also coming in 6's table.
So, it is also a factor.
24 is also coming in 6's table.
So, it is completely divided by 6.
So, it is also a factor.
So, what is the hcf of these 3? 6.
Now, we will learn how to find the LCM of any number.
I hope, you have understood the hcf.
We will also learn the hcf of big numbers.
Now, let us start with small numbers.
Because we will do it from small number only, then only we
will be able to tackle big numbers easily.
So let's see, what we do for this question?
Let's find out LCM, that is 6, 18 and 24.
What we have to find out for 24?
LCM, find their LCM.
LCM is Least Common Multiple.
So we have to run table of 6, 18 and 24.
So we have to find out a multiple which is in table of 3.
So we have to find out a number which is in table of 3.
And what is that number? It should be the smallest number.
That will be our LCM.
So look at 6.
6 becomes 2 into 3.
18 becomes 2 into 3 into 3.
24 becomes 2 into 2 into 2 into.
So what we do for HCF question?
In HCF question, we look at least power.
Like we did question earlier.
So what did we see? 2, 2 was there.
3, 3.
In all these, what is the least power of 3?
In this, 3 power is 1.
In this, 3 power is square.
In this also, 3 power is 1.
So least power of 3 is 1 in common 3.
So we took that in HCF.
What was the least power of 2 in common 3?
In this, 2 power is 1.
In this, 2 power is 3.
So what is the least power? 2 power is 1.
So what do we take in HCF?
In HCF, we take least power.
Least common power numbers.
But in LCM, we have to take highest power.
If you see, 2 power is 1 and 3 power is 1.
What is this? 2 power is 1 and 3 power is 2.
What is this? 2 power is 3 and 3 power is 1.
This is 2 power is 1 and 3 power is 1.
3 power is 1.
So what do we have to do?
This is the least common multiple in name.
But when we find out from this method, what do we find out?
We take highest power of common factor.
Like in this, what is the highest power of 2 in 3?
What is the highest power of 2 in 3?
So we will take 2 power is 3.
What is the highest power of 3 in 3? 3 power is 3 square.
So we will take 3 square in 3.
Suppose we have some other number.
Suppose we have 6 into 5.
So what is the prime factorization of 6 into 5? 2 into 3
into 5.
And what happens in 3? 2 into 3 into 5.
So 5 is an extra number.
5 is only in this.
It is not in these two.
It is not in 18 and 24.
It is only in 5 into 6 that is 30.
So what do we do with 5?
Will we take it or not?
Now I have told you that we have to take highest power.
If 5 power is 1, then we can also write 5 power is 0.
5 power is 0.
1 is the factor of all.
So 5 power is 0 and 1.
So what is the highest power of 5? 5 power is 1.
So we have to multiply that also.
Indirectly, if we do not have any common number.
Like 5 was not common in both.
But in LCM, we have to take that also.
So what is this?
LCM of 30.
So LCM of 30, 18 and 24.
What do we have? 2 power 3 into 3 square into 5.
8 into 9 into 5.
That is 40 into 9.
We will get 360.
Now 360 is a number which will come in table of 30, 18 and
24.
This is the smallest number of all.
Which is common in these three tables.
So this is LCM of 30, 18 and 24.
If we take the previous question, LCM of 6, 18 and 24.
What was that?
It did not have 5.
In that, only 2 power 3 and 3 power 2 were coming.
So it becomes 8 into 9.
That is 72.
Did you understand? Now you will try some questions on this.
How to get LCM and HCF of any number?
Now these numbers were natural numbers.
They were natural numbers.
There was nothing different in them.
Suppose now we have,
Instead of natural numbers,
If we have fractions or decimals, Then how will we find out
LCM and HCF of them?
So first of all, you write.
First of all, we will write fractions.
Because it is comparatively easier than decimal.
So this was the first case.
That how we are doing it with natural numbers.
So now we will see,
LCM and HCF of fractions.
How will we have fractions?
It will be in the form of P by Q.
Means it will be in the form of numerator and denominator.
Suppose we have to find out LCM.
How will we find out LCM of any fraction?
Suppose we have some numbers in fraction.
Suppose we take 2 by 7, 3 by 5 and 5 by 8.
We have taken these three fractions.
2 by 7, 3 by 5 and 5 by 8.
Now we have to find out LCM.
So how will we find out LCM? We don't have to do anything.
We have learnt LCM and HCF of natural numbers.
Now we have to proceed with them.
What will we do in such fractions? We are taking LCM.
So what will we find out in numerator?
LCM.
Now how will we find out LCM? Whatever terms are there in
the numerator.
Means 2, 3 and 5.
If we have to find out LCM, then we have written LCM above.
Below it, we have to write HCF.
What will we find out HCF of?
Of the denominator terms.
That is 7, 5 and 8.
So this will be found out.
LCM of this fraction.
So solve it.
What will happen?
LCM 2, 3 and 5.
What is common in 2, 3 and 5?
We will find out LCM like this also.
With one more method.
See this long division method.
2, 3 and 5.
Keep on dividing with prime factors.
Till all 3 are there.
Below it, 1 should not be there.
So if we divide 2 with 2, then 1 is there.
If 2 will not be there with 3, then write 3 as it is.
Write 3 as it is.
If 5 will not be there, then write 5 as it is.
Now 1 is there.
Now how will we divide 3 with 3? 3 ones are 3.
And 5 as it is.
Now we will divide 5 with 1, 1, 1.
Till 1, 1 does not come in the end.
Till then we carry forward.
So what is the LCM we have? 2 into 3 into 5.
So what is the LCM of 2, 3 and 5? 2 into 3 into 5.
Now we have to find out HCF of 7, 5 and 8.
Now we have to find out factors.
What will be the factors of 7? We can write 7 as 1 into 7.
We can write 5 as 1 into 5.
We can write 8 as 2 power 3.
2 into 2 into 2.
So is there any common factor?
Except 1, there is no common factor.
So what will be the LCM of 7, 5 and 8?
Its HCF will be 1.
Because there is no common factor in 7, 5 and 8.
So what will be the LCM of 7, 5 and 8? 7 5 or 8 1 so this
answer will come 6 into 5 that is 30 by 1 that is 30 so we
have this fraction of LCM that will come 30 now if we have
to find HCF then what will we do in HCF first you do this
much LCM for HCF
we will do like we have to find out LCM so we were keeping
LCM above similarly we have to find out HCF so what will we
take above
HCF of numerator whatever terms are coming in numerator
their HCF and in denominator what will we do LCM of
denominator whatever terms are there
do this question with the help of question let us see find
HCF of 3 by 7, 6 by 21, 12 by 12 by 40 what you have to
find out is HCF so what will be the HCF of any fraction
HCF will be of 3 by 7, 6 by 21 and 12 by 14 if we are
taking out HCF then what will be above
HCF will be there
HCF of numerator terms 3, 6 and 12 and what will be in
divide HCF will be upside down that is LCM 7, 21 and 14
LCM of denominator terms whatever we have to find out of
numerator terms and whatever is left in denominator take
out the other thing like if we have to find out HCF then
take out LCM so see HCF of 3, 6 and 12 so we will write 3's
factors what will be
the factors of 3 6 will be 2 into 3 so 12 will be 2 into 2
into 3 so we have to find out least power see 2 is not
common in all 3 so 2 will not come 3 is common in all 3 so
what will be the common factor of 3 so the common factor
which is 1 will be highest
so what will be the HCF of 3 now we have to find out LCM we
will find out LCM from long division method means from
division method so what will be 7, 21 and 14 first of all 3
are going from table of 7 so this will be 7 so 1, 3 and 2
then 2 3 is going
from 2 as it is then 3 now all 3 are 1 so here stop so this
is their LCM that is 7 into 2 is 14 14 into 3 14, 3 is a
that is 12, 1, 3, 1, 3 42 so we have 42 of this
LCM of 7, 21 and 14 so what is the HCF of these 3 fractions
3 upon 42 this is our answer did you understand how to find
out fractions now I will tell you that if we we learned
this method in LCM division method to find out LCM suppose
if we have to find
out HCF with the help of this suppose if we have to find
out HCF with the help of this then how will we find out so
let us see how will we find out HCF with the help of this
HCF with the help of which with the help of division method
so just now we did 7, 21 and 14 so first of all divide this
with the prime number which is common in these fractions so
what is common in these fractions 7 is seen, so we are
dividing with 7
so whatever number these 3 are dividing with write that
first so I wrote 7 so what is this 1, 3, 2 but from which
these 3 are dividing, no so we have solve this as it is 3,
this is 1, this is 2 this is 1 this will cancel out from 2,
then this is 1 so
what was the common number in this common was only 7,
common factor only 7 was the number with which these 3 were
dividing so what will be 7 its HCF
HCF 7, 21 and 14
and what will be the rest of the LCM
LCM will be the multiplication of these 3 like what we did
in the last class 7 x 6 42 became LCM so with the help of
division method how will we find out HCF so what we have to
do we will do the common factor once we do for big number
how will we find
out HCF suppose we are doing 60, 28 and 34 this is what we
have what we have to find out for this
HCF using division method so 60 first write all 3 60, 28,
34 so first of all these 3 are even numbers so these 3 will
cancel out from 2 so what is the common number 2 how will
it go from 2 2 x 30 is 60 2 x 14 is 28 and 2 x 17 is this
take any other number
34, we are getting only one take 36 36 so what will we get
36 so in 36 we will get 18 2 x 18 is 36 why did I take 36
because it will become common to tell you a little more now
again what is it going from again these 3 are going from 2
so from 2 15 x
7 and what we have is 18 now there is no common number 7 is
going alone from 7 so 7 is not common so it becomes 1, 9
and 15 then these 2 will go from 3 so it will become 5, 1
and 3 then it will go from 3 it will become 1, 1 and what
will go last from
5 all 3 will become 1 so what is common in this green part
in this part I have written it is common what are the 3
numbers cancel out so what will be the hcf hcf will be 2 x
2 that is 4 so what will be the hcf of these 3 we have 4
means the factor highest
common factor we have 4 once you try this with that method
normally we have done with prime factorization method once
you try with that and see if 4 is coming and what will be
the lcm lcm will be the product of all these if we do the
product of all these
numbers then it will be lcm so try once with that method do
all of them till then I will not see you do yours then see
here and tally so what will happen we could write it as 6 x
10 then split it that is 2 x 3 x 2 x 5 so we can write 60
as 2 square x
3 x 5
I hope it is visible this color is visible
I think no so let's do it with yellow color ok now how can
you write 28 what can you write 28 7 x 4
2 x 2 x 7 then 36 36 is 6 square 6 x 6 that is 2 x 2 x
2 power 2 into 3 power 2.
You have done this, you have prime factorized them.
Now see what is the common factor in them.
See this, if we make this 2 into 2 instead of 2 square.
And make this also 2 into 2 instead of 2 square.
Okay, so what is common with them? One 2 is common, see.
Second 2 is also becoming common.
And some are becoming common.
3 is coming in this, 3 is coming in this.
But 3 is not coming in 28.
7 is coming only in 28.
Not coming in other two.
Similarly 5 is also coming only in 60.
Not coming in other two.
So what is their common? 2, one 2 came from this one. One 2
from this.
So 2 into 2.
What is 4? Highest common factor.
So 4 is ours.
It also came with the help of this.
Now it is verified that what will happen with both methods.
It will be same.
This is the choice of the student.
Which method will be easily available in the exam.
Or which method has the fastest speed.
Speed also matters along with your knowledge in such exams.
Because all these are time limited.
There is a time foundation.
To submit them.
Otherwise someone can ask this question for a day or two.
But that is not the case.
We have to learn time management properly.
And solve them with speed and accurately.
So this is done.
We have HCF of fractions.
Normal and natural numbers.
Next we are moving on to.
Finding LCM and HCF of decimals.
How will we find out decimals?
Third is.
HCF and LCM of decimals.
Suppose we have a question.
We learn with the help of question.
Whatever decimals we have.
What will we do first?
First we will make the digits after the decimal.
We will write in like terms.
Suppose we have three things.
Suppose I write the question.
We have a question.
That find the HCF and LCM of 3, 0.9 and 0.12.
3, 0.9 and 0.12.
We have to find out HCF and LCM.
So the first step is.
After the decimal, the number of digits.
Whatever numbers we have.
We will make it equal.
We will make it like terms.
After the decimal, there is not even one.
After the decimal, there is one digit.
And after the decimal, there are two digits.
So what is the highest?
If it is of two digits.
So what to do after the decimal?
Put two digits.
How can we write 3?
We can write 3 as 3.00.
It will be the same.
We can write 0.9 as 0.90.
After the decimal, you can put any number of zeros.
It will not make any difference.
If it is stopping here, 0.9.
If the decimal is terminating at 9.
Then you can put any number of zeros.
It will not make any difference.
How can you write 0.12? 0.12 has two digits after the
decimal.
So now there are two digits in all three.
Now what to do?
Remove the decimal.
After 2, there is a decimal.
So remove the decimal.
Now solve without the decimal.
If we remove the decimal, it will become three handed.
0.90 will become 90.
0.12 will become 12.
So what we have to find out?
We will find out LCM and HCF of these.
Then find LCM and HCF in a simple way.
Then whatever result will come.
In that, leave two digits.
Means, leave the last digit and the last second digit.
Because after two digits, there is a decimal.
So leave two digits in it and put a decimal from the last.
So what will happen to us?
Their LCM and HCF.
Did you understand so much?
So see 30, 90 and 12.
Use any method to find out their LCM and HCF.
Let's find out HCF first.
So first we will find out HCF of 300, 90 and 12.
Whose LCM and HCF we have to find out?
HCF of 300, 90 and.
So how can we write 300?
3 into 100.
Then divide 100 more.
How can we write 100? 4 into 25.
We can write 4 as 2 into 2.
And we can write 25 as 5 into 5.
So if we prime factorize 300.
Then 3 into 2 square into 5 square.
We have written 300.
How can we write 90?
We can write 90 as 3 square into 10.
Then split 10 into 2 into 5.
So 2 into 3 square into 5.
25 into 4.
Now how can we write 12?
We can write 12 as 2 into 6.
That is 2 into 2 into 3.
So what is the highest common in this?
We can see 2.
1 power 2 is common.
So 2 is the common factor.
So first we have 1 factor 2.
If we write least power.
Then 2 power 1 is least.
So it becomes 2.
Now what is the least power of 3? 3 also has least power.
We have 1.
This has 2.
This also has 1.
So what is the power of 3?
That is 1.
Now what is the least power of 5?
This has 2.
This has 1.
And this has 0 power of 5.
It means there is no power of 5.
So there is nothing in this.
So we do not have 5 common.
So what is the common factor of these three?
We have 6.
Common factor is highest.
So we have highest common factor of these three.
6.
Now we have 6.
I told you that whatever is the highest common factor i.
What we have to do?
Whatever we have made after digit decimal.
We have to leave that in this.
So if we want 2 decimal.
So we put 0 here.
Put another 0 here.
So leave 2 here.
And what do we put here?
Decimal.
So what will be the result of 0.06?
Hcf of 3, 0.9 and 0.12.
Similarly, you can do LCM also.
You just have to do nothing.
You have to find out the result of 390 and 12.
Let's find out that also.
So we have to find out LCM.
We have to find out LCM of 390 and 12.
First of all, cut it from 3.
3, 100, 30 and 4.
Then what is it going to be common? 2, 50, 15 and 2.
Now it is not going to be 2.
So common we have.
We can also get common from this method.
So this is the common factor.
And what was our answer?
It was 6.
So it was 6.
Now let's solve this.
How will it go?
First, let's cut it from 2.
So 1, 15 as it is.
And it will be 25 instead of 50.
Now how will it cancel out?
From 5.
So in table of 5, 5, 3 and 1.
If I do this with 5, then 1, 3, 1 and the last 3, so 3, as
it is.
So, what will be the LCM? 3 x 2 x 2 x 5 x 5 x 3.
That is 3 square x 2 square x 5 square.
So, you have 9 x 4 x 25.
25 x 4, 25 x 4 will be 100.
And 100 x 9 will be 900.
So, 900 is the LCM of 390 x 12.
But, what we have to do? We have to find the decimal.
So, after this, leave 2 places and then put the decimal.
So, 9.00 will be what? 9.00 will be the LCM of 3, 0.9 and
0.12.
So, what will be the LCM of these? 9.00 and...
So, what did we learn in today's class?
We learned to find the LCM and HCF of any natural number.
And then we learned the fractions.
And in the last, we learned the decimals.
So, keep practicing this much.
Next, we will learn the properties of LCM and HCF and their
related questions.
Thank you.
Count me in. I'll make sure to spread the word to all my
classmate.