Hello everyone, welcome to another video on Quantitative
Aptitude.
In the last video, we all studied about the number system.
The subcategories of the number system, i.e. complex number
and further subcategories of real and imaginary part.
And further subcategories of the real and imaginary part.
So, first of all, where did we start from?
We started from the natural number and from the whole
number.
Then we studied about prime and composite.
So, today we will study about natural number and whole
number.
So, if we multiply some natural numbers or whole numbers,
what will be the result?
So, we will learn some particular names of those numbers.
Then we will understand their meaning and we will learn the
formula to find out those numbers.
So, let's start.
So, first of all we have written factors and multiples.
In your childhood, you must have added and subtracted the
natural numbers.
And you must have learned tables.
How did you learn tables?
By multiplying two numbers.
So, today we will find out these factors and multiples.
We will do this for numbers.
Now, we will not study the factors of polynomials.
We will study them later.
Today, we will keep our video restricted to numbers only.
So, people get confused between factors and multiples.
What is the difference between these two? Suppose, if we
are asked about factors, people say multiple.
If we are asked about multiples, people say factors.
So, they are in this dilemma.
So, today we will understand what is the particular
difference between these two.
Suppose, we have studied tables.
In your childhood, everyone has studied tables.
So, here you have written tables of 2, 3, 4 and 5.
Everyone knows this.
This is written here.
2 x 1 is 2, 2 x 2 is 4 and so on.
So, we get numbers by multiplying two numbers.
For example, if we multiply 2 by other numbers in tables of
2, we get 2, 4, 6, 10 and so on.
So, we get numbers by multiplying.
So, what will happen?
Multiply word is related to multiples.
So, what will happen to these side numbers?
Multiples.
You have told multiples.
But, what will be the particular multiples of this number?
These will be multiples of the number which is coming in
the table.
For example, all these numbers are coming in the table of 2.
So, these will be multiples of 2.
Similarly, if we see the table of 3, 3, 6, 9, 12 and so on.
All these numbers are being generated by multiplying 3 by
other natural numbers.
So, all these numbers which are generated on the right hand
side, these will be multiples of 3.
Because, these are coming in the table of 3.
So, what will happen to these?
These will be multiples of 3.
Now, see the speciality of multiples.
What are the multiples?
The multiples of 3 are coming in the table of 3.
How are we generating them? 3 into some other natural
number.
So, if I want to write multiples of 3, So, can I write
their general form? We are multiplying 3 by any natural
number.
So, let us suppose we call it as K.
So, if we multiply 3 by K, K will be any other number.
So, what will happen to this?
This will be multiples of 3.
That is, this will come in the table of 3.
Similarly, you can also do multiples of 4 and 5.
So, see the multiples.
How many multiples do you have in total? How many multiples
do you have in total?
Here, it is given 4 x 12, 48, 5 x 12, 60.
So, will you have only 12 multiples? Will you have only 12
multiples?
No, you can have an infinite number of multiples of any
particular number.
Because, this table will continue.
If we multiply 3 by any odd number, we will get even bigger
multiples.
And this will continue.
Because, what are natural numbers?
They are infinite.
They keep on moving.
Similarly, what will be the multiples of any particular
number?
Infinite.
There will be infinite number of multiples.
Infinite number of multiples.
Whose? Of any particular number.
Of any particular number.
Which form are multiples?
It can be any number.
Suppose, we have a capital number N.
We have to tell the multiples of this in which form?
So, what will be the multiples of this?
The multiples of this will be in the form of N.
That means, if we multiply N by any other number, then the
number which will come after multiplying both of them, will
give us the multiples of N.
So, this will become an infinite number.
Now, we have learnt the multiples.
The second word we have is factor.
What will be the factors?
Like this multiple, whatever came on the right hand side,
we had a multiple.
The things which are on the right hand side, Suppose, we
have taken the number 16.
See, what are the multiples of 16?
We have 16.
2 into 8, that is 16.
So, 2 and 8, whichever numbers are being multiplied to
generate this number, we will call these numbers as
factors. Whose factors will we call?
Obviously, the factors which are being multiplied, will be
the factors of 16.
Now, how do we tell the factors?
There is a form of factors.
As we have seen, there is a form of multiples of N.
So, there is a form of factors as well.
In which form do we write any particular factor?
What will we do with any particular factor? What will we do
with this 8? We will split it.
In which form will we split it? We will split it in the
form of prime numbers.
Suppose, this is 2.
How will we prime factorize it?
We will prime factorize it by writing it in the form of
prime numbers.
For example, this is 8.
What will we do with 8? We will divide it by 2.
First of all, the number 2 is the prime number.
We will divide 4 by 2.
We will get 2.
Then, 2 times 1 is 2.
How can we write 8?
We can write 8 as 2 times 2 times 2.
We will get 8.
So, instead of 8, we will get 16.
16 is equal to 2 times 2.
Instead of 8, we will get 3 times 2.
So, all these are the factors of 16.
What will be the factors of 16? The factors of 16 are 2.
It is coming in the table of 2.
In the table of 1, every number comes.
So, the factors of every number must be 1.
Because, all the numbers come in the table of 1.
So, it is 1.
2 is also coming.
So, it is 2.
Now, multiply these two.
What is coming? 2 times 2 is 4.
So, what is 4?
It is the factor of 16.
Look here.
We have seen the table of 4.
4 times 4 is 16.
Can you see it?
So, 4 is also the factor of 16.
Now, multiply all three.
2 times 2 is 4.
4 times 2 is 8.
So, 16 is also coming in the table of 8.
So, 8 is also the factor of 16.
Now, we will multiply all.
16 will come.
So, 16 comes in the table of itself.
So, 16 is also the factor of 16.
So, what are the total factors of 16? 1, 2, 4, 8 and 16.
So, what are we getting?
We are getting a finite quantity.
Finite means, which we can count.
Which can have a particular value.
So, we can calculate the total.
How many total numbers of factors did we get? 1, 2, 3, 4, 5.
So, how many total factors of 16 did we get?
We got 5 total factors of 16.
Whereas, if we were looking at the multiple of any number.
How were we getting that?
Infinite.
So, what is the difference between multiples and factors?
Factors are finite in number.
And multiples are infinite in number.
So, this difference is clear.
So, let me write the definition once.
Let me explain what are factors and multiples?
Suppose we have two numbers, one is n and the other is m.
We are saying that n divides m.
What does it mean?
It means that n is completely dividing m.
m is a number and n is completely dividing it.
If we divide m by n, then we get the remainder as 0.
This means that it fully divides m.
So when will this be the case?
When will it completely divide?
When m comes in the table of n.
When m comes in the table of n, then only it will be able
to divide it completely.
So here m becomes multiple of n.
And this n which is dividing m, will become a factor.
Whose number?
Whose number is dividing m?
So this will become a factor of m.
Did everyone understand this much?
What is a factor?
It is the number which is being written in the tables of
the numbers.
For example, if we want to find the factor of 4.
What is the factor of 4?
It comes in the table of 2.
We can write it as 2x2 or 4x1.
So it comes in the table of 1, 2 and 4.
So the number which comes in the table of any number,
becomes a factor.
And the final result which is getting multiplied in the
table, becomes a multiple.
Is it clear to everyone?
Repeat this definition in your mind.
See if you can understand what are multiples and factors.
How many of any particular number?
Are you able to understand?
So now we will see how to find the factors and multiples of
any number.
Let's see.
Factors and multiples of any particular number.
Suppose we want to find the factors.
What number did we take?
We took 16, so let's take 25.
We have to find the factors of 25.
So how will we find the factors?
We will find the factors with the help of Prime
Factorization Method.
What is Prime Factorization?
We had studied the prime numbers in the category of number
system.
What are prime numbers?
They are the numbers which do not come in the table of any
number, except in the table of 1 and itself.
And apart from 1 and itself, they do not come in the table
of any other number.
They are our prime numbers.
What is Prime Factorization?
Factor means to multiply them.
So what can we do with any number?
We can write it in the multiples of prime numbers.
Any number can be expressed as a product of prime numbers.
We can express any number as a product of prime numbers.
Or we can write it.
Suppose this is 25.
How will we factorize it?
This is 25.
What comes first from 2? 2 comes first.
Then 3, then 5, then 7.
These are all prime numbers.
Then 11 and so on.
So first we will check with 2.
It does not have divisibility with 2.
Because it is an odd number.
So it will not go with 2.
It does not have divisibility with 3.
You can read all these in divisibility theory.
Which number is divisible by which number or not.
How to find it out?
That video will also come soon.
So it will not be divisible by 3.
Because it does not come in the table of 3.
24 comes in the table of 3.
But 25 does not come.
So it is not divisible by 3 or 2.
So what is left?
Now what comes in the table of 5? 5 times 5 is 25.
So 5 will cancel out.
So 5 times 5 is 25.
5 times 5 is 25.
Now 5 is a prime number.
So it will cancel out from 5.
Until 1 comes below 5.
You will keep running the process.
So we can write 25 as 5 times 5.
Or we can write it as 5 square.
Multiplying 2 times.
So what will happen?
Power will be added.
1 plus 1 is 2.
So what will happen? 5 times 5 is 25.
Multiplying 2 times.
So what will happen? 5 times 5 is 25.
So what are the factors of 25? 1 is 5.
And 1 will come.
1 always comes.
1 is a factor in factors.
1 factor is 5.
And 5 times 5 is 25.
So what is 25? 1 is 25.
These are factors of 25.
Similarly, you can find out any number using factorization
method.
Using prime factorization method.
By representing it in terms of prime.
So these are the factors of 25.
Now suppose, how many factors are there? 1, 2, 3.
So how many factors are there in any number?
How many total numbers of factors are there?
Can we find out this also using any formula?
Sure.
There is one formula using which we can find out how many
number of factors are there in a particular number.
Okay.
So let's note down that formula.
Suppose we have a number capital N.
This is the number.
Now we have seen that we can factorize any number in prime.
We can factorize any number in prime.
Suppose we can factorize any number in prime.
So see, power is also varying.
5 can be square.
We have seen the case of 16.
2 power is 4.
So power can be 4.
6 can be 6.
2 into 3.
So power is 1.
So power can be anything.
So I have given the name of that power.
So this is P.
Now this is 1 prime.
2nd prime in multiply is B.
2nd prime is B.
So power of B is Q.
Similarly, 3rd prime can also be power of R.
Similarly, there can be more prime.
As many as we want.
So it will go on.
Now I am telling about 3.
So what is there in this?
Where A, B, C.
We have prime numbers.
And what is PQR?
What is PQR?
Natural number.
First of all, we will convert any number in this form.
We will factorize it in prime.
Then we will take it out.
One minute.
Then we will take it out.
Write the formula.
Total number of factors.
Of N We are finding out.
So this formula is
We don't have to do anything.
We have to do this.
The powers.
What are the powers we have?
We have PQR.
We have to add 1 to them.
So what will happen?
P plus 1 into Q plus 1 into R plus 1
Whatever the powers will be.
We have to add 1 to them.
What do we have to do?
We have to multiply them.
We call this formula
Tau N The one which we are taking out of this number.
Write it in bracket.
Tau is a representation.
Generally what happens is
In your course, there are many formulas.
So those formulas We remember those formulas.
If we remember their particular name.
Then what happens to us?
There are chances of remembering.
So what is this?
Total number of factors.
Of N
That is P plus 1 into Q plus 1 into R plus 1
As we apply in this.
This was 25.
What were the total factors? 5 What will we do in 5?
What is the value of A? 5 What is the value of P? 2 So what
will happen?
P plus 1
That is 2 plus 1 2 plus 1 is 3.
2 plus 1 is 3.
So how many numbers were there? There is no Q.
Because there is no prime number, so there is no Q.
There is no R.
So only this will work. P plus 1. We have got that 3.
And what was the answer of the total number of factors? 3
was there.
Okay.
So we will understand that the total number of factors from
which formula we will subtract from any number.
This is the formula of the total number of factors.
Now we have subtracted the total number of factors from any
number.
Next what we have is that these factors are finite numbers.
So suppose we are asked what will be the total sum of these
factors?
So how will we subtract that? We will see the formula of
that.
Now what is there? We can only subtract the sum of the
factors.
Because the multiples will become infinite.
It will go in series. If we do the sum of the infinite
series.
So now what we will learn? We will learn to subtract the
sum of the factors.
So let us see.
Suppose first of all, if we have any number n.
We will write it in this form.
A power P into B power Q into C power R.
Where A, B, C are prime.
And P, Q, R are natural numbers.
So we have prime factorized any number.
Then what we have to do?
Let us see the formula of sum.
Sum of all the factors of.
Okay.
What will be the formula of this?
Do this.
Start A with zero power.
Then increase the power of A by one by one.
So after zero, what will be the power? A power will be one.
Then what will be the power?
A power will be two.
While increasing the power, how much power we have to
increase? The number which we have been given.
Suppose if we have A power five.
Then what we will do? A power zero plus A power one.
Plus A square plus A cube plus A power four.
Plus A power five.
How much will this work?
Whatever power P is given to us, this sum will work till
there.
So what will happen?
Last, A power P.
Then what we have to multiply in this?
We will do the same of the second number.
Like we did of A.
We will do the same with B.
We will also start B with power zero.
Plus B power one.
Plus B square.
Plus, how much will this work?
It will work till Q.
So B power Q.
Then multiply.
Then third number.
Third prime number.
We will also start with zero.
C power zero.
Plus C power one.
Plus C square.
Plus up to C power eight.
So what will happen to us? Sum of all the factors of N.
It was up to C.
There were three prime numbers.
So we did three multiply.
If there were four, we would have done the fourth one too.
And so on.
We can do as many as there are.
So suppose we had taken an example.
What number did we take?
Twenty five.
We took the number twenty five.
What was twenty five?
The number twenty five that we have.
How could we write it?
Five square.
We could write five square.
The factors of this, I had told you.
What were they?
One comes in the table of one.
One comes in the table of five.
And one comes in the table of twenty five.
So what do we have to do?
What do we have to do for all these factors?
We have to sum.
That is one plus five plus twenty five.
So if we are summing, what is coming?
See, calculate it yourself.
Thirty one is coming.
If we apply it in this formula,
So let's verify,
Is it satisfying?
So the power of A.
We have A as five.
And we have P as two.
So we have to apply the power of A only up to two.
So what will happen?
The sum will happen.
The power of A is zero.
That is the power of five is zero.
Plus the power of five is one.
Plus the power of five is two.
We will apply it only up to two.
Because what is the value of P?
Two.
So we will apply it up to two.
Now the power of any number is zero.
It is one.
So it is one.
Plus five.
Plus five square.
That is twenty five.
So see, this is coming.
This is also coming.
Thirty one.
So both are giving the same result.
So what is this formula?
It is verified.
So you must have understood this much in factors and
multiples.
We will do more categories of numbers.
That is, we will do prime numbers, co-prime.
We will do HCF and LCM further.
With their help, we will learn to read co-prime.
We will read twin prime and so on.
So thank you for this video.
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