Hello everyone, I hope you have moved ahead in this video
after watching the last video, so in the last video we did
factors and multiples
We will move ahead of them, the numbers that will come to
us after moving ahead, what will we do with their help?
Categorize, first of all, the numbers that are coming in
the numbers ahead, when we taught you the natural numbers,
so what was the subcategory of them, you had odd number,
even number and prime number and composite number.
What were the prime numbers whose factors were only
themselves and theirs was 1.
There should be no other factors other than 1 and itself.
So this will be the prime number.
What will be the composite number? If it has 1 and any
other factor other than itself, then that number will be
called composite number.
So now we will see one more category, what will we have?
What will we have? Twin prime number.
So you write new thing that is twin, twin.
Now as it is coming in prime and twin, what is twin?
You must have heard the word twin, twins.
But what will be the meaning of twin prime?
Like the first number that should be there, we will check
them first.
Are they prime or not?
First of all, that number should be prime.
Now what will be the additional condition added in twin
prime?
That number should be consecutive prime numbers.
It should be consecutive.
Consecutive means one after another.
If we have 2, 3, 5, 7, 11, these are our prime numbers and
so on.
So see, consecutive means first 2 is coming, then just next
to it is 3.
So 2 and 3 will be consecutive numbers.
2 and 5, similarly these two are also coming one after
another.
So 3 and 5, sorry not 2, 3 and 5.
These two will also be consecutive prime numbers.
Similarly 7 and 11, after 7 11 is coming.
So what is this? Consecutive prime numbers.
So what do we need for twin prime?
First of all, those consecutive prime numbers should be
there.
Plus, what will be the additional condition added in this?
That their difference should be only of 1 and 2.
What should be the difference in both?
Difference of 2 should be there.
Like we saw 2 and 3, these are consecutive prime numbers.
But is their difference 2?
If we see their difference 3-2, then what will happen?
It will be 1, there is no difference of 2.
So what will happen to these two?
They will not be twin prime numbers.
Similarly, if we see next tuple of 3, 5.
What should be the first condition of these two?
That consecutive prime numbers should be there.
Now see their difference.
What will be the difference? 5-3 that is 2.
So their difference is 2.
So what will happen to this tuple?
It will be an example of twin prime.
Example of twin prime.
Now similarly see 5 and 7.
5 and 7 are also consecutive prime numbers.
And their difference is also 2.
So what will happen to 5, 7?
We have twin prime numbers.
7 and 11 will not be there.
Similarly, we can also see in the next tuple.
Which will be there and which will not be there.
So this is the first category.
That is twin prime number.
Second category is related to factors.
That will be your second category.
Co-prime numbers.
You must have heard of co-prime numbers.
Co-prime numbers.
Now what is this?
Twin prime or co-prime.
This is not for a particular number.
We see their relation.
Means we see the relation between more than one number.
Then we say that it is twin prime or co-prime.
So this is one particular number.
Like prime number.
We used to say only one particular number.
Is it prime or not?
Composite number was also one particular number.
But in this we have more than one number.
So what will be co-prime?
We have learned factors.
So we will call two numbers or more than two numbers as
co-prime numbers.
If and only if they don't have any common factor except one.
Means they don't have any other common factor except one.
So what will happen?
No common factor other than one.
One is the factor of all.
Which is same in both numbers.
Like if we see.
And that number is not necessary.
In this name is coming prime.
But that number is not necessary.
That those two numbers should be prime only.
That non-prime composite number can be anything.
We just have to see that their common factor.
We should not get any other name except one.
Suppose I took.
Suppose I took 3.
And next number I take 8.
3 and 8.
We have to check these two numbers.
So what will be the factors of 3?
Factors of 3?
Factors of 3 will be 1.
1 comes in 3 3 table.
There will be no other factors.
What will be the factor of 8? 8 comes in 1 table.
8 comes in 2 table.
8 comes in 4 table.
And 8 comes in its own table.
So how many factors we got in total of 8?
We got total factors of 8 as 4.
And 3 got 2.
So see what is common in these two? Common is only 1 in
both.
Rest of the numbers are not common.
This has 3.
This does not have 3.
This has 2, 4, 8.
This does not have.
So how we got 3 and 8? These two numbers became co-prime.
So in these two factors common is only 1.
Rest of the numbers are not common.
So these are the co-prime numbers.
Next we have perfect numbers.
Suppose we have any number n.
We will get some factors of n.
Suppose we got 1 factor of n.
A, B and C.
We got these 3 factors.
Now when we will sum all these factors.
So what will happen?
First of all its factor will be 1.
It will be 1.
And n will also be its factor.
Every number has its own factor of 1.
It comes in its own table.
Apart from that suppose their factors are A, B and C.
So we have to sum all these except n.
Apart from n, we have to sum all the factors of n.
If we will see their sum.
Like we saw 1 plus A plus B plus C.
So if it comes equal to capital N.
Then that number will be called perfect number.
So that number will be called perfect number.
This is the first example.
The easiest example is 6.
How can we write 6?
We can write 2 into 3.
2 into 3 is equal to 6.
And what are the factors of 6?
It comes in the table of 1, 2, 3 and 6.
So which number do we have to sum? 1, 2, 3.
We will not take 6.
We will not take that number.
Otherwise the sum will exceed that number.
So 1, 2 plus 3.
1 plus 2 plus 3.
This is also 6.
So it itself comes equal to the number.
So what is the number 6?
It is a perfect number.
One example of perfect number is 6.
Try the next example of 28.
Take out the factors of 28.
1 and 28 will be there.
And the remaining 2 and other factors.
What will you do with them? the properties If we sum the
remaining factors, we will get the same result.
Next thing if we see that if we add all the factors of n in
reciprocal way and this time we will not exclude n.
If we add all the factors in reciprocal way, then we will
get 2.
This is the second property of perfect number.
Like we saw 6, 6 was our perfect number.
We had 6 as perfect number i.e.
2 x 3
What were the factors of 6? 1, 2, 3 and itself.
Now I said that we have to sum all the factors in
reciprocal way.
So we should get 2.
If we do reciprocal of 1, then it will be 1 x 1.
If we add all the factors in reciprocal way, then it will
be 1.
First take LCM. I will teach you LCM later.
But now see this.
Everyone's LCM will be 6.
So it will be 6.
If we want to make it 6, then we have to divide 6 by 6.
If we want to make it 6 by 3, then we have to divide 6 by 2.
And this 6 is already there, that is 1.
Now if we sum them, then it will be 12 x 6.
And if we divide 12 by 6, then it will be 2.
So the sum of all these is 2.
So this is the second property of perfect number.
Now I told you that I will tell you about co-prime.
So what is new for co-prime?
New thing for co-prime is that suppose you have been asked
to tell us all the pairs of co-prime which are up to 10.
Means how many pairs or tuples of co-prime will be formed
in 10 numbers.
So how will we find out that? We will find out that as we
have to take up to 10.
So we will factorize 10 as prime.
That is 10 will be 2 x 5.
We have factorized 10.
Then what we have to do?
Then we have to write 10.
In that we will do 1 minus reciprocal of first factor.
1 minus 1 by 2.
In this we don't have to take 1 and 10.
Only these factors will come, that we have to take.
So 1 minus 1 by 2 x 1 minus 1 by 5.
So what is this?
If we solve this, then it will come as half.
This will come as 4 by 5.
After solving this, how much is this? 10 minus 2 by 5 will
come as 4.
So in 1 to 10, the number of co-prime pairs which we have,
how many will be there?
Total 4 pairs will be formed.
Suppose you are asked, how many co-primes will be there up
to 50?
So what will you do?
First you will factorize 50.
Then you will apply this formula.
So you write this formula.
Number of co-primes between 1 to N.
From 1 to N, how many co-prime pairs will be there? So what
will be this formula?
First of all, we have factorized N.
How did we write N?
A power P, B power Q, C power R.
So what we have to do is, N into formula N.
Multiply N.
1 minus reciprocal of A into 1 minus reciprocal of B into 1
minus reciprocal of C.
This will be our formula.
We just studied, In factors, we named a formula as Tau N.
So we have the name of this formula.
Phi N.
Capital N.
Whatever we have to find out, we wrote it in bracket.
This is called Euler's function.
I hope you have understood this much.
These are also subcategories.
This number was left in the system.
Because this was related to factors.
So we have done this.
We will do this in next class.
We will learn LCM, HCF.
Thank you for watching.
I will make sure to spread the word to all my knowns.