Hello, welcome back to another video on LCM and HCF.
In the last videos, we learned how to calculate LCM and HCF
for any two or three numbers, fractions and decimals.
And before that we studied factors and multiples.
So, in today's video, we are going to do properties and
results based on LCM and HCF.
So, the first result or rule that is made from LCM and HCF,
that is, if we are given two numbers,
LCM of those two numbers and HCF of those two numbers will
be given.
So, the product of those two numbers, that is, the product
of their LCM and HCF will be equal to the product of those
two numbers.
So, this is our first rule that the product of two numbers,
if they are two numbers, A and B, then their product means
we have to multiply them. So, what will we get A into B?
Is equal to their LCM, whatever is the LCM of A and B, into
their HCF, whatever is the HCF of A and B.
So, let's see a question related to this.
Suppose we have a question, product of two co-prime numbers
is 117.
Product of two co-prime numbers is 117.
You can mark the important information like this.
Then their LCM is, we have to find out their LCM.
So, what is given to us?
The product of any two numbers is given and how are those
two numbers?
Co-prime.
When we have been called a co-prime number, what did we
mean by that?
In the previous video, when we did factors and multiples,
that when will there be co-prime two numbers?
When they will not have any common factor, only one common
factor will be there, that is 1.
We will not get any other common factor other than 1.
So, when we will not get any other common factor other than
1, then the highest will also be the same.
So, the indirectly given co-prime here, it means indirectly
that what will be the HCF of these two numbers? 1.
What will be the highest common factor of these two
numbers? 1.
Now, what is given for the product of those two numbers?
117.
So, let us suppose we take those two numbers A and B.
A and B, let two numbers be A and B.
So, what is given?
A into B is equal to 117 and their HCF.
HCF of A and B is 1.
So, we have to find out LCM.
Now, see in this property, if you put the value of the
product, the product of two numbers is given, then 117 will
be equal to?
LCM of A and B, we have to find out.
Into HCF.
HCF is 1.
So, what is the LCM? If we divide by 1, we will get the
same number.
So, what will be the LCM of A and B in this question? 117.
So, what is the answer?
Answer is 117.
That is LCM.
Did you understand this question?
I hope it would have been easy.
It would not have been asked directly.
There will be some more advanced questions of this type.
What will we do gradually?
We will solve it in the next lectures.
Next property will be your rule number 2.
Note down rule number 1.
So, what will be your rule number 2?
Let us see one more question related to rule number 2.
Look at this.
The HCF and LCM of two numbers are 12 and 924.
HCF is given.
LCM is given of two numbers.
Let us suppose two numbers A and B.
Let us assume two numbers A and B.
What else is given?
Their HCF is said.
HCF of A and B is equal to 12.
And LCM of A and B is 924.
Then find the number of such pairs.
HCF and LCM are given.
So, we will get the product of those two numbers.
That is we will get A into B.
What will we get A into B?
HCF into LCM.
That is 12 into 924.
We will do 12 into 924 quickly.
924 into 12.
8, 4 and 18.
Into 924.
We got 88011.
We got 11088.
What are the two numbers whose product we got?
We got 11088.
We have to find out two such numbers.
To find out two such numbers, we are given that their HCF
is 12.
HCF is the highest common factor.
Highest common factor means the factor of those two numbers
A and B.
One common factor of these two numbers is 12.
There can be other multiples also.
But the common factor is 12.
So, can we write A and B in such a way that A will be the
multiple of 12.
Suppose 12x will be the multiple of 12.
And B will be the multiple of 12y.
Y and X are the other constants.
Why did we take 12?
Because the highest common factor of these two numbers is
12.
So, they will be the multiple of 12.
And the rest of them will be the multiple of X and Y.
We will not find any other common factor in these two
numbers.
That is why 12 is the highest common factor.
If the value of A is 12x and the value of B is 12y.
Then we will put A and B in this equation.
The value of A and B is 12x into 12y is equal to 11088.
So, X and Y will be 11088 divided by 12.
If we solve this equation, what will happen? 6, 2, 6, 2, 5,
5, 4, 4.
Then 6, 2, 7, 7, 7 and 2.
Then 3, 1, 3, then 16, 8, then 12, 6.
Then again, this is 3, this is 6, 18, 9 and 3.
Then 3, 1, 3, 3, 2, 3, 1, 3, 1, 3, 21, 7 and 21.
Then 21 that is 7.
So, what is the product of X and Y? We have X and Y.
I hope it is visible.
We have X and Y.
Multiply it and get 77.
Now, we have to find such a pair.
In which we have to multiply it and get 77.
So, what should we do?
We indirectly subtract the factors of 77.
So, how can we write 77? We can write it as 1 into 77.
1 into 77.
And how can we write the other one?
We can write 7 into 11.
Are there any other numbers in these two products which we
can multiply and get 77?
Are there any other numbers?
So, what are our two pairs? One is 1, 77 and the other is
7, 11.
We got these two pairs whose multiplication is 77.
And it satisfies the given things in this question.
So, what will be the answer of this question?
We had to find such pairs.
So, this will be the answer.
We are getting two pairs which satisfy these two pairs.
Now, let's move to the next rule.
So, the next rule is rule no. 2.
What does rule no.
2 say?
Suppose, we have given the ratio of two numbers.
You must be knowing the ratio.
If we divide any number by a fraction, So, if we write it
in lowest form, then we can write the remaining two numbers
in fraction form also.
Suppose we have 4 upon 8, this is one fraction.
So, how will we write it in lowest form?
It will be cut as 2 4 ones and 4 twos.
So, write it as 1 by 2 or 1 is to 2.
This is the ratio form.
This is the fraction form in lowest form.
And this is the original fraction form.
So, if the ratio of two numbers is A is to B.
If the ratio of two numbers is A is to B.
Like here the ratio of 4 is to 8 is 1 is to 2.
Similarly, the ratio should be in lowest form.
So, the ratio of two numbers will be multiple.
Like if we multiply 1 by 4, then we will get 4.
And if we multiply 2 by 4, then we will get 8.
So, we are cutting out the common thing.
Similarly, if we multiply the common thing in lowest form.
Then will we get the original number?
Yes, we will get it.
Because we are cutting out the common part and making it
lowest form.
And you must have seen this in the question of ratios.
Let the common ratio be X.
If we are given any ratio, suppose 3 is to 7.
So, what we do is, let the common ratio be X.
Then we divide it by 3X and divide it by 7X.
This is the common part which we cut out to make it lowest
form.
So, if we have two numbers whose ratio is A is to B.
So, in this case, what will be those two numbers?
A into K will be one number.
And the other number will be B into K.
What will be K in this case?
K will be a constant.
The common part which we divide will be a constant.
So, what will be those two numbers?
A into K will be one number.
And the other number will be B into K.
Now, if we get these two numbers.
And this was in the lowest form.
A is to B.
It means that nothing was common in these two numbers.
Because this is in the lowest form.
If it was common, then it would have been cut out.
So, nothing is common in these two numbers.
If two numbers are of A and B.
Then what will be the HCF in this?
What is the HCF?
Highest common factor.
So, A and B are not common in this.
Only K is common in both.
So, in this case, what will be the HCF of these two numbers?
We have K.
This is also in the result.
If we have this in the lowest ratio.
Then we can find out those two numbers.
Of A and B.
And what will be their HCF?
We have K.
Similarly, we will have LCM of these three products.
That is, A and B.
Both of them were different things.
In which nothing was common.
A and B.
And K is common.
So, we will write the common thing in LCM only once.
So, what will be this?
A, B, K.
We will have LCM of A, B, K.
And HCF will be only K.
Okay? Let's see this question.
If we have two numbers.
4 by 8.
We don't know what are those two numbers.
What is the best?
Their common ratio is 1 is to 2.
Suppose.
1 is to 2.
So, what will be those two numbers?
Those two numbers will be either K into 1.
Or 2K.
And K will be any constant.
Then we will subtract LCM.
Okay? Next, moving on to rule number 3.
What will be rule number 3?
Rule number 3 says.
Suppose there is a number X.
We are dividing it by three different numbers.
For example, we are dividing X.
We are dividing X by A, B and C.
We are dividing X by these three numbers.
When we are dividing X by A.
Then we have remainder R.
When we are dividing X by B.
Then also we have remainder R.
When we are dividing X by C.
Then also we have remainder R.
It means that after dividing by three numbers.
We are getting same remainder.
So, how can we find out X?
The value of X will be.
X will be equal to K plus R.
What is K here?
And what is R?
You can see R.
The notation which we have already used.
That is remainder.
The same remainder which will be left after dividing by
three numbers.
That will be R.
And what will be K?
K will be LCM of A, B.
So, what will be K?
LCM of A, B and C.
So, this rule is based on LCM.
LCM is being used in this.
So, this is done.
What is the value of X?
K plus R.
Now, let us see related question.
Try to try it once.
With the help of the same formula and rule.
Let us see.
The least number which when divided by 4, 4, 6, 8, 12 and
16.
And least is also important.
Least number which when divided by these numbers.
Leaves a remainder.
Leaves a remainder 2 in each case.
2 in each case means.
If we divide that number by these five numbers.
What will we get in each case?
Same remainder that is 2.
So, what we have to find out?
We have to find out that least number.
That X we have to find in this question.
So, what should we do?
It was written simply that that number will be X.
That is K plus R.
And what will be R?
Remainder which was same for all.
Its value is 2.
This is R.
So, instead of R keep 2.
Now, what was K?
K was our LCM of all these numbers.
That is 4, 6, 8, 12 and 16.
This will be the value of X.
So, we have taken out LCM of 4, 6, 8, 12 and 16.
4, 6, 8, 12 and 16.
So, first of all from which is it going?
From 2.
2 is a 3 is a 4.
2, 6 is a 12 and 2, 8 is a 16.
Again 3 is going from 2.
2, 1.
3 is not going.
2 is a 3 is a and 4 is a.
Then 2 is going from 2.
3 is not going.
1 is a 3 and 2.
Again 2.
So, here 1 came.
3 as it is will remain.
Now, it is going from 3.
So, 1, 1.
1, 1 has come.
What should I do here?
Stop.
So, what LCM we have? The product of all these is 2 into 2.
2 into 2.
Power of 2 is 4.
Into 3 plus 2.
I have put this in the formula of LCM.
So, what is this? 16 into 3.
16 into 3.
What will you have?
That is 20, 18.
And 3 into 3 is 48.
48 plus 2.
So, what will be the value of X? 50.
So, what is the answer of this?
Simple.
We should know the rule.
So, we can easily solve this question.
Otherwise, you would have done the whole of HCF LCM.
Then you would have put the concept.
In that way.
Then it would have been time taking.
And what we have to do? We have to do the time management
in the exam.
We have to solve it in speed.
So, we will solve it like this.
Write it.
Next, we will move to rule no. 4.
In question no. 4, the remainder of 3 is opposite to the
remainder of 3.
For example, in question no. 3, we had the same remainder.
That is, after dividing by 3 numbers, the remainder that
was remaining was same.
But if we get the case that the remainder of 3 is
different, then how will we solve that question?
How will we find out the value of x in that?
So, there is a rule for that. Write it down.
Rule no. 4 is also based on LCM.
So, what will happen in rule no.
If we divide x by 3 numbers, a, b, c,
So, if we are dividing by a, the remainder is p.
If we are dividing by b, the remainder is q.
If we are dividing by c, the remainder is r.
What is this? Remainder.
Remainder when a, b, c are divided by x.
So, this will be p, q and r.
So, what will be the value of x in this?
In the last one, we did plus.
In this, we will do minus.
Before doing minus, we will note down one more condition.
That we should have one more condition.
That a-p, b-q and c-r should be equal.
Only then we can apply this rule.
If these three conditions are equal,
Then what will be the value of x?
It will be k minus t.
Now, neither t is being used nor k.
So, what will be the value of t?
Whatever we are writing,
These three things will be equal.
We will make it equal to t.
These three things are equal.
That is the value of t.
In the last question, we saw that k is the LCM of a, b and
c.
LCM of these three numbers.
So, k will be the LCM of a, b and c.
And what is t?
These three values will be equal.
We have t.
So, let's do a question on this.
Here is the question.
What is the question?
Read the question quickly.
The question is,
A number which when divided by 10,
A number,
We take that number as x.
Which we have to find out.
A number which when divided by 10,
Dividing by 10,
Leaves a remainder 9.
When we are dividing x by 10,
What is the remainder we are left with? 9.
When divided by 9,
Leaves a remainder 8.
When we are dividing by 9,
We are left with 8.
When divided by 8,
Leaves a remainder 7.
When we are dividing x by 8,
What is the remainder we are left with? 7.
We have to find out x.
If we look at the rule,
What is the value of a? a is 10.
b is 9.
c is 8.
We have got the values of all three.
Remainders, Remainders of 1 were given the name p.
So, p is 9.
q is 8.
r is 7.
We have got a, b, c, p, q, r.
So, we can find out x.
What is the value of x?
We have got k minus t.
But, we forgot to check the condition on which we had to
apply the second condition.
Let us check whether it is fulfilling or not.
Only then we will do this.
So, what was the condition? a minus p, b minus q, c minus
r. All three should be equal.
So, if we do a minus p, 10 minus 9 is 1.
If we do b minus q, 9 minus 8 is also 1.
c minus r is 8 minus 7.
This is also 1.
All three are equal.
So, we can use this formula for the equal values.
Now, when the equal values are obtained, What was the equal
value of?
It was equal to t.
So, instead of t, we will put 1.
And what was k?
What was the LCM of? a, b, c.
That is 10, 9 and 8.
So, k is LCM of 10, 9 and 8.
And what is the value of t? It is 1.
Put the value.
LCM of 8, 9, 10.
What will come?
All three are different.
There is nothing common in them.
So, if we do with 2, then we get the product of them. But,
I will show you.
So, 2, 5, 4, 9 as it is.
Then 2, 2, 9 and 5.
Then 2 again.
How will 9 go from 3 into 3? So, 3 into 3, 5 will also go
from 5.
So, if we multiply all these, will it come? 8 into 3. 8
into 9 into 5.
So, what do we have?
We did not get the product of them.
We did not get 10 because 10 and 8 had a common, 2. That is
why we got the product of them.
8 into 9 into 5.
That common 2 was multiplied once.
So, what will come? 8 into 5, 40.
40 into 9, what will come? 360.
So, we got the product of LCM. 360.
And what is the value of T? 1.
So, 360 minus 1.
So, what is the value of X? 3, 5.
So, this is our answer.
The rule number 1 was easy.
This rule and the other rule.
We just have to remember what we have to get.
How will it come?
It had K plus R.
When we have the remainder of all three.
It has K minus T.
When the remainder is different.
And what is the value of T?
A minus remainder.
What was left from it?
B minus B remainder.
C minus C remainder.
The equal value of all three will come.
That is equal to T.
Got it?
Let's move ahead.
It is not getting too boring.
Remember this rule.
It is getting too long.
But when you memorize it.
You will solve all the questions.
Which depend on this.
Which are based on this.
Got it? Next, we have rule number 5.
Rule number 5 and rule number 6.
Which will depend on?
HCF.
Like we had.
Rule number 3 and rule number 4.
Depend on LCM.
Similarly, rule number 5 and rule number 6.
Depend on?
HCF.
It is the same question.
It is the same type.
It is like X.
We had divided X.
It is similar to rule number 3.
If we are dividing X.
From A, B, C.
Sorry.
If we are dividing ABC.
From X.
Then we have the same remainder.
Same remainder.
And that is R.
Same remainder.
R is coming.
If we were asked.
The type of question.
In that, your number was coming.
Least.
Find the least number.
In this, what we have to find out?
As a greatest search X.
In that, we have to find out least X.
Or say least number.
Or any other number.
What is mentioned in this?
In this, it will be mentioned.
That we have to find out the greatest search X.
So, how we will find out?
We will find out it.
With the help of HCF.
Now, how we will find out with the help of HCF?
What we have to take HCF for? We have to take HCF.
X will be the value.
HCF of.
A minus B.
B minus C.
B. and C-A.
The difference between these three numbers, if we take each
pair, first we will subtract A from B, then B-C, then C-A.
The three numbers that we get from these three numbers, if
we subtract their Hcf, then it will be equal to
X.
This was easy, there is nothing to do in this.
Now, because in this remainder, what is the main thing?
It should be the same.
What should be asked in the question?
The greatest should be asked.
And then what we will do?
We will subtract the Hcf of the difference.
So, what will happen to X?
We will get S-X.
If we see the second case of this, as we saw in that, that
if the remainders are different, then what will happen to
us?
Then how will we solve it?
So, let us see that rule once, that if we have different
remainders.
So, rule no.
Y was X, we are finding out greatest such X.
So, this is done.
Again, Hcf rule.
So, A, B, C are being divided.
Remainders are same.
P is obtained from this,
Q is obtained from this, and R is obtained from this.
What are all these?
Remainders that we will get.
Means, in this case, we do not have same remainders.
We have different remainders along with all three.
So, in this, we will find out the value of X.
So, in this, the value of X will be, X will be Hcf.
Then we will subtract Hcf because it is based on Hcf.
We had taken the difference of numbers on that.
What will we do in this?
We will do, Do you remember rule no.
We had taken A-P,
A-Q, B-Q,
C-R and it should have been same.
So, this is also similar.
In this, what we will do?
A-P, B-Q, and C-R.
These three numbers that we will get, we will solve and
take their value.
So, that will be the value of X.
That would be greatest such
X.
Now, what we will do on this?
Let's see the question.
See the question on this.
Try to try the question by reading the rule.
If you are not able to understand, then read the rule again.
So, the greatest number by which,
First of all, the greatest number came.
So, which rule should we have?
The following rule no. 5 and 6.
The greatest number by which 7, 9, 2 This is A.
and 9, 0, 1
This is B.
are divided.
We need such number which will be divided by this.
Leaving the remainder 9 and 5. In that, we will get 9 and 5
respectively.
That means, if we divide X by this, then what we will get?
We will get 9 remainder.
and what we will get in that? 5.
So, this is the value of P and this is the value of Q. We
have already written A and B.
So, what would be X in this case?
X would be HCF of
A minus P B minus Q C is not there.
So, it is on two numbers only.
So, we will run it till two.
HCF of A minus P and B minus Q
So, what will we get
A minus P?
A minus P is 7, 2, 9 minus 9 7, 2, 9 minus 9 and B minus Q
is 9, 0, 1 minus 5
So, we will solve this and get
HCF of 7, 20 and 8, 9, 6
What we have to find out for this?
HCF Whatever will be their HCF, that will be the value of X.
So, should I find out HCF here? Can you see it?
Let's find out HCF.
7, 20 and 8, 9, 6 First of all, let's see from 2.
So, what is this? 3 is a 3 is a Then, what we have to see
When we take out HCF from division method, then the common
factor from which these two are getting divided, we will
multiply it from there only. So, from what are the common
factors getting divided?
Now, it is from 2.
2, 3 is a 6 and 6, 12, 6 is a and 0, 360 Then, 4 is a 4 and
16 that is 8. Then, again it is from 2 because they are
even.
So, we will divide it from 2.
1, 16 8 is a 0
This is 2 to 4.
Then, 2 is common.
So, 2, 90 and 1, 1 Then, again it is from 2.
Usually, you divide it from 3 or 11 because it is a square
of 11.
But, now we are seeing common numbers, common factors.
That's why we will divide it from 2 because 2 is common
from both.
So, this will be 45.
And, what will be this? 5 and 12 that is 56. Now, is there
any common number which divides 45 and 56? No, there is no
common number other than 1.
2 is not there because it is odd.
3 will not be there because 3, 5 plus 6 is 11. So, it will
not go from 3.
In 4's table, again 45 is not there. Similarly, in 5's
table, this is not there.
In 6's table, also 45 is not there.
So, we don't need to solve this further. Because, we only
need common factors.
So, for HCF, what are the common factors?
That is 2 to the power 4.
And, what is 2 to the power 4?
We have 16.
So, what will be the answer?
The greatest X which divides from 7 to 9, 9 to 1 and gives
remainder 9 and 5, that is 16.
So, the answer is 16.
I hope you all understood.
HCF and LCM have the same rules.
And, the type of questions will vary further. I have just
asked you about properties and their questions. But, these
questions will come in some other type where
LCM and HCF will be used.
So, we will do that in the further video where we will
solve the questions.
Thank you.
No, I haven't.
What's it about? Basically, if you refer the graduation
BVOC program to your friends and they sign up using your
referral code, you'll get some awesome rewards.
That sounds interesting.
For every friend you refer, you earn points that you can
redeem for discounts or even cash rewards and also get the
job guarantee after completing course. Count me in.
I'll make sure to spread the word to all my classmates.