Hello everyone, welcome back to our new video.
I hope your practice is going well on number system and
your questions are being solved easily.
And you are practicing advanced questions.
What we are going to do in this video is Remainder Theorem.
If we are doing theory of anything, we should know
something about it.
The four most important things we should know in this is
that we should know the remainder.
What is the remainder of anything? When does it come?
When will we get the remainder?
When we divide two things, the remainder we get is the
remainder.
For example, if we have a fraction, when will we divide two
things?
Either write in fraction or divide.
Suppose we have a question, that is 14 divided by 6.
How will we write 14 divided by 6 in fraction?
We can also write 14 by 6 in fraction.
14 is numerator and 6 is denominator.
If we want to divide it, how do we divide it?
We divide it like this.
We will write 14 inside and 6 here.
Now everyone knows that 6 twos are 12.
How much we have to count 6's table? As much as we have
this number or any number smaller than this.
For example, 6 ones are 6, 6 twos are 12.
12 is smaller than this.
As soon as we see 6 threes are, it will become 18.
If it is bigger than this, we will not go to 6 threes.
We will stop at 6 twos.
So 6 twos are 12.
We will write 12 here.
Then we will subtract these two.
So 14 minus 12, we will have 2.
So what we have left below, we will call it remainder.
We will call this remainder.
This 2 above, we will call it quotient.
This part in numerator, which we wrote inside, we will call
it dividend.
And in denominator, which we wrote outside, we will call it
divisor.
All these terms should be clear to you.
It is clear now.
Suppose we have 14.
If there is any number above 14, we will carry it out below
and solve further.
Now see, whatever remainder is left below,
Whatever is left, it will always be divisor.
Means, what will we get from the one we are dividing?
It will always be smaller than that.
It will be always less than the divisor.
Why? Because if it is equal to the divisor,
Or bigger than that,
Then we will add the table in front of it and then divide
it.
Suppose we had remainder 6 here.
When would we get remainder 6?
When we would add 6 more to it.
If we would add 6 more,
Then what would we get? 18 18 is fully divided by 6.
So whenever we have remainder,
It will always be smaller than this.
It will never be equal to divisor.
So, note down all these.
Remainder will always be less than.
Less than.
We will say less than.
We will not even say equal to.
Only less than.
It will always be less than the divisor.
One thing is that,
Remainder will always be less than the divisor.
Second thing is,
If we see, 6 into 2 is 12.
6 into 2 is 12.
If we add remainder in 12,
We will get 12 plus 2 is 14.
Which is equal to dividend.
The second property of this is,
It is very important.
That is,
Dividend.
The dividend which we are dividing,
It will always be equal to,
Divisor into quotient plus remainder.
Remainder is written in short form.
So, Remainder.
Dividend will be equal to,
Divisor into quotient plus remainder.
Dividend is 14.
14 is equal to 6 into 2 is 12 plus 2.
This is 40.
Both are equal.
Left hand side is equal to right hand side.
This is the basic knowledge of remainder.
We have studied this in small class.
The thing which you have not studied,
We have that.
Negative remainder.
What is negative remainder?
Negative remainder.
Whenever we write remainder,
What we do is,
In the last,
If anything comes,
I said,
It will be a small number.
Either take the table to the equal number of the dividend.
Or take the table to the small number.
So, what will be the remainder?
It will be positive.
So, where did the negative remainder come from?
Where did the negative remainder come from?
Let's see that.
Suppose, 6 into 2 is equal to 12.
So, the remainder is positive 2.
But, what can we say in another way?
We can say that 14 is small.
If we reduce 2 into 14,
Then it will be divided.
But, if we go to a higher table,
For example, 6 into 3.
What will 6 into 3 be? 6 into 3 is 18.
So, what is the relation between 18 and 14?
What will that be?
If we subtract 14 from 18,
Then what will we get?
Minus 4.
So, then what do we get?
The remainder becomes negative.
If we go to a higher table,
When we go to a lower table,
For example, we stopped at 12.
Then we got a positive remainder.
If we go to a higher table,
For example, if we stop at 18,
Then what will we get? 14 minus 18.
That is, minus 4.
So, in that case,
What will we get?
Negative remainder.
Most of the time, we use this since childhood.
But, if we want to solve big questions quickly,
Then the negative remainder also plays an important role.
So, if I show you the negative remainder,
Then what will we do for the negative remainder?
For example, take this.
What did we have? 14 by 6.
This is what we had.
So, if we go ahead in 14,
What will we get?
We will get the divisible of this.
So, what will happen?
Just after 14,
How much will we have to subtract in 14 by 18?
Then what will we get?
Then we will get the remainder.
We got the remainder of this.
Now, we saw 2.
This is your positive remainder.
And when we went to 18,
We got the same remainder.
Minus 4.
So, that is your negative remainder.
Did you understand the difference
Between positive and negative remainder?
I hope you understood.
Next, we will study some properties.
Suppose we have two numbers.
We have to find out the remainder.
a plus b divided by n.
We have three numbers.
In the numerator above,
We have the plus sign.
And in the denominator,
We have the common number.
That is n.
So, we will find out the remainder
By adding these two numbers.
Which will be equal to which?
Either we will add and divide.
And we will get the remainder.
Or we can write it separately.
Remainder of a by n.
We will divide a by n.
And see what is the remainder.
Then we have the plus sign in the middle.
So, we will get the remainder.
b by n means the one which will come after dividing b like
we take this example it is 14 by 6 instead of 14 by 6 if we
have been given a question if we change it in this form so
let me write 14 like this suppose I write 12 plus 2 by 6
what we have to
find out for this we have to find out the remainder so if
we apply the same property then what will happen if we
divide the remainder 12 by 6 plus remainder 2 by 6 now 12
by 6 12 is being fully divided by 6 so what will be the
remainder? 0 because it
is being fully divided
so we get 0 plus remainder 2 by 6 now when we write 2 2 is
already smaller than 6 so 6 will be 0 by 0 so 2 minus 0
will be 2 if we have a smaller number than 6 then we will
get the same remainder like any number from 1 to 5 any
number from 1 to 5 or 0
to 5 any number from 0 to 5 in the numerator if we have
divisor 6 then we will get the same number as it is so now
we have remainder 2 so 0 plus 2 is 2 so this remainder when
we split 14 we got 2 and when we divided 14 we got positive
remainder 2 so you
can use any method the answer will be the same now suppose
I do 14 by 6 14 by 6 14 by 6 we can convert it to lowest
form and divide it we are dividing it by 2 so it will be 2
times 7 so we will get 7 divided by 3 we will divide it so
if we do 7 divided
by 3 7 3 times 2 is 6 so this is the remainder so what is
the remainder 1 but did we get 14 by 6 remainder 1 in that
we got remainder 2 so here we are wrong so why we are wrong
we have solved it process wise in lowest form so this is
wrong because dividing
this is same in lowest form but in divide we will get
different remainder so to get same remainder what we will
do whatever we have taken common and divided we have to
multiply it like we took 2 common and divided so whatever
remainder we got in that
we have to multiply that common number then we will get
answer so when we do 1 times 2 so the term will be the
answer of remainder this will not be the answer so you
understood the difference suppose i give you another
question we have 125 divided by
5 125 divided by 5 so if we see it this is fully divided
this is fully divided this is fully divided this is fully
divided we will see some other question 48 divided by 5 48
divided by 5 solve this 48 divided by 5 48 divided by 48
divided by 15 what will
happen 48 will go inside 15 will come out 15 will become
divisor 48 will become dividend 15 table till 48 15 times 1
is 15 15 times 2 is 30 15 times 3 is 45 15 times 3 is less
15 times 3 is 3 15 times 3 is less 15 times 3 is 3 15 times
3 is 3 15 times
3 is 3 but if we see 48 and 15 what is the common thing 3
times 5 is 15 3 times 1 is 6 3 times 16 is 48 3 times 16 is
48 so this is 16 divided by 5 if we take remainder of 16
divided by 5 16 and 5 we get 1 we get 1 but remainder
should be 3 so I told
you cancel out any common factor cancel out any common
factor 3 multiply 3 by remainder 3 times 1 is 3 which is
our actual answer this problem is also clear next we will
do some theorems based on remainder based on remainder so
first theorem is
Fermat's theorem this is your Fermat's theorem so what is
Fermat's theorem suppose we have a number suppose we have a
positive number and we have a prime number suppose we have
a positive number and we have a prime number so if we want
to find out remainder
we have to find out a power p minus 1 divided by p so we
get we get 1 but there is another condition between a and p
there should be a relation that is a and p should be
co-prime co-prime to each other co-prime to each other
means there is no common factor
there is no common factor indirectly we get hcf of a and p
should be 1 if we have these three conditions that is p
prime, a positive number and hcf of a and p is 1 if we have
these three conditions we will get this type of question a
to the power p minus
1 divided by p what is the remainder there is no remainder
there is only 1 if I tell you another corollary suppose
suppose we are asked suppose suppose we have we have to
find out a to the power p divided by p we have to find out
its remainder now I will
tell you a property if we have addition in the numerator in
the numerator if we have addition in the numerator in the
numerator in the numerator in the numerator in the
numerator in the numerator Similarly,
multiplication also has a property. If in our numerator,
suppose two numbers are multiplying, and we have to find
the remainder of them by n. So, what will happen in this
also? In this also, it will split up. And in the middle,
there was a plus sign,
so there was a plus sign. In this, there is a multiply
sign, so there will be a multiply sign in the middle.
So, what will happen? First, I will find the remainder of a
by n.
Then, I will multiply the remainder of b by n.
Whatever is the remainder, I will divide it by b by n.
What should I do with these two things?
I will multiply them with each other.
So, if we do the same thing here, then what result do we
know till now? Till now, we only know the result of the
Fermat's theorem.
And we know the result of the remainder.
So, what will we do in this? We will do, we will take a to
the power p.
In this, we are given all the conditions.
That p is prime, a and p are co-prime, a is a positive
number.
This is already given to us.
But this time, instead of a to the power p minus 1, a to
the power p is asked.
So, what will we do? We will split it in this.
I can write a to the power p as a into a to the power p
minus 1.
I can write it like this, right? And there is p in the
divide.
What do we have to find out for this?
We have to find out the remainder.
We have done this.
Because the power is in the multiply.
If these two are in the multiply, then what will be the
power?
It will be added.
Power 1 plus p minus 1.
So, p will remain as power.
So, we have converted this into this.
Now, this is in the multiply.
So, do it separately.
Like, by applying this rule, what will happen?
Remainder a to the power p into remainder a to the power p
minus 1 by p.
So, this can be applied in this.
Fermat's theorem will be applied in this.
So, this will be 1.
And remainder a to the power p.
What will be the remainder of a to the power p?
It will be a only if it is less than a.
If it is less than a to the power p, then what will be the
remainder?
We will get a only.
So, we will get this.
Remainder a.
And what was this?
Remainder 1.
So, a into 1 is a.
So, we have a new corollary from Fermat's theorem.
If we write this new property.
If we are asked about remainder a to the power p by p.
Then what will we get?
We will get a.
So, we got to know two things from Fermat's theorem.
One is this result.
This is our main result.
If you are not able to remember this, then it is okay.
You can use this help.
But you have to remember this.
So, let's do a question on this.
Let's see the question on this.
The question is.
Find the remainder of 250 to the power 42 divided by 41.
First of all, see the denominator 41.
What number do you see? 41.
You can see the prime.
41 is a prime number.
So, what is 41?
It is a prime number.
That is, we call it p.
What is the value of p? 41.
See, what is above? 250.
250 and 41. 41.
Is there anything common in these two? If there is
something common in these two, then it is the
multiplication of 41.
Because 41 is a prime number.
There are no other factors.
So, 250 is not coming in the table of 41.
So, what is the HCF of these two? We have 1 of 250 and 41.
That is, these two have become co-prime numbers.
So, what is 250?
A.
So, 41 and 250 have become co-prime.
A, 250 has become a positive number.
P has become a prime number.
So, all three conditions have been fulfilled.
So, what should I apply in this?
I should apply Fermat's theorem.
Now, what was in Fermat's theorem?
Either we had power p minus 1 or power p.
But, what is the power in this? 41 plus 1.
That is 42.
That is, p plus 1 is the power.
So, how will we solve this?
Look at this.
Split this.
We can write the power of 250 as 42.
The power of 250 is 40 plus 2.
We have split 42 into 40 plus 2 divided by 41.
Why did we do it in 40 plus 2?
Because we need one power of p less for Fermat's theorem.
That is, we need p minus 1 in power.
So, p minus 1 should be 40.
That is why I did this.
40 plus 2.
Now, when the power is in the add, it will be multiplied.
So, we will get the power of 250 as 40 into the power of
250 as 2 divided by 41.
Now, we have to find out the remainder of this.
Now, what happened? These two things are in the multiply.
So, the remainder will also be different.
It will be distributed.
And there will be a multiplication sign in the middle.
So, this would be the remainder of 250 power 40 divided by
41 into the remainder of 250 power 2 divided by 41.
Now, look at this.
This looks like Fermat's theorem.
We have p minus 1 as 41 and p minus 1 as 40.
What will be the remainder of this using Fermat's theorem?
The remainder of this will be 1.
And what will be the remainder of this?
Split the square power again.
That is, the remainder of 250 power 41 into the remainder
of 250 power 41.
So, solve it once and multiply it twice.
It is square.
So, 250 power 41.
Divide 250 power 41 by 250 power 41.
So, look at this.
Once 41 is 6.
What will be the remainder of 41 power 6?
That is 246.
So, 246 is 41 power 6.
If we subtract, what will be the remainder? The remainder
will be 4.
Remainder will be 4.
So, write it here.
1 into 4 into 4 is 16.
So, we will get 16.
Remainder would be 16.
And this would be our answer.
Suppose we get a bigger number by multiplying this
remainder.
That is, a bigger number than 41.
So, we divide it by 41 again and write the remainder.
Because the remainder can never be greater than divisor.
If it is greater than divisor, then divide it again.
Then write the answer of the final remainder.
Is it clear to everyone?
There should be no doubt in this.
Till now, we have studied the properties.
We are dividing.
And we have studied only one basic theorem.
That is Fermat's theorem.
The next theorem on remainder is Wilson's theorem.
What does Wilson's theorem say?
Before doing Wilson's theorem, I would like to tell you a
notation.
That is factorial of a number.
I have not told you this.
So, let me clear it once.
Suppose we are asked the factorial of a number.
A factorial of any number.
First of all, you see what will be the sign of the
factorial.
We will have the sign of the factorial.
The sign of the factorial will be like this.
Exclamatory mark that.
Suppose we are asked 15 factorial.
We are asked 15 factorial.
We write factorial like this.
So, factorial means, as we are asked 15 factorial.
So, we have to start from 15.
And we have to multiply it by decreasing it.
Till 1.
If we subtract 1 from 15, it will be 14.
So, 15 into 14.
Then we subtract 1 into 13.
And so on, it will go on till 3 into 2 into 1.
That means, we have to multiply it till 1.
Either you say it like this.
Or you say it like this.
So, this is the definition of factorial, then we will do
Wilson's theorem, so Wilson's theorem will give us the
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We have already learned about Wilson's theorem.
Now we are going to learn about some important points.
Suppose we have this type of term.
Similarly, a to the power n minus b to the power n.
Suppose we have n odd.
A to the power n minus b to the power n is divisible by a
minus b.
When n is odd.
When n is even.
We will get this factor.
We will get a minus b factor.
We will get one more factor.
That is a plus b.
That is a plus b and a minus b.
If we have n even.
Then we do not have any factor.
Note this down.
Next point is.
Suppose we have this type of term.
The power of numerator in this term is some number.
Some number is n.
Then what will happen?
Then what will be the remainder of this term?
Remainder of this term would be.
It will be k to the power n.
The power of n will go here.
Why will this happen?
If we divide this term.
We can also multiply this term.
Because this term will multiply n times.
Because this term is power n.
If we break this term with addition.
Then we will get two terms.
x upon a plus k by a.
This is the remainder of this term.
Similarly, we will get n times.
This is the same thing.
I am telling you one packet.
Similarly, you can do n times.
So, what is this total? n times.
This is the remainder.
Then this remainder is also distributed in this.
So, ax by a.
In this, a by a is being fully divided.
We are getting a factor above.
So, it will be cut.
So, the remainder of this will be 0.
Because the term above is being cut by a.
If we multiply x by a.
Then we will get ax.
So, the remainder is 0.
Similarly, what will happen to k by a?
The remainder of k by a will be k.
So, the total is 0 plus k.
The final answer is k.
The remainder of 1 is k.
So, if we multiply this by n times.
Then we will get k times.
We can write k times as k power n.
Can we write?
Now, we have the corollary of this.
Suppose we get the value of k as 1.
Then what will happen?
If we get the value of k as 1.
Then we do not have to do anything.
Then we have to put 1 instead of k.
And the answer will be the same.
So, this becomes a new.
The remainder of ax plus 1 divided by n.
Divided by a.
And the power of the above should be n.
We will get the remainder of 1.
Because we will power anything of 1.
The power of n should also come.
But if we power anything of 1.
Then we will get the answer as 1.
So, this becomes ax plus 1 to the power n.
Divided by the remainder of a.
If we see the same.
If we get minus instead of plus.
What will be the place of plus?
Minus k.
Then we do not have to do anything.
We will replace k with minus k.
So, next we will have.
Next we will have.
Remainder of ax minus k to the power n.
Divided by a.
Its remainder will be minus k to the power n.
Minus k to the power n.
Now, what will be the corollary according to this?
If we have k1.
In this case.
In the second case.
Then what will happen?
Minus 1 to the power n.
It will give us a different answer.
So, what will happen?
Remainder.
This is its corollary.
That is remainder.
Ax minus 1 to the power n.
Divided by a.
What will happen?
It will have two values.
One will be minus 1.
And the other will be 1.
Why? Because its answer will be minus 1 to the power n.
Minus 1 to the power n.
Can give two possible values.
That is, it can give minus 1.
When we have n.
What will happen?
If we have n, it will be odd.
And it will give 1 value.
If we have n.
Power even.
So, whether you do this or not.
You do this.
With their help, you will also get this.
So, if I give you one more thing.
I will show you one more corollary.
You remember this.
One minute.
Half a minute.
Memorize it.
I will tell you one more corollary.
What can be the next corollary?
Suppose.
Next corollary is related to that.
Related to the same type.
Suppose we have.
a to the power n.
Divided by a minus 1.
What is asked for this?
The same remainder is asked.
What will be the remainder?
So, how will we do this?
You see, power n is coming.
And in the last also.
In this one also, power n was there.
So, in this type.
What will we do?
We will reduce it.
How can we reduce it in this type?
Look at this.
Whatever was below.
The same a was above.
Multiplied by x.
We can write it like this.
We have to bring above also.
a minus 1.
So, we write it.
If we put minus 1 plus 1 in it.
Then there is no problem.
Because minus 1 plus 1.
What will it give?
It will give zero.
So, it is not bringing any effect or change.
So, we do it.
a minus 1 plus 1.
And this whole power n is left.
Whole power n is there.
Divided by a minus 1.
What we have to find out?
Remainder.
These two things are same.
So, you see.
a minus 1.
This term and this is one.
Both should be same.
The a which was above x.
The same a was below.
So, suppose if we take value of x in this.
1.
Means a minus 1. one key power n.
So in this question what will be the answer? 1.
Irrespective of what we have, if n is odd and n is even,
for all n.
So what can you write in other language? For all n.
For any n.
For any n this will be true.
Similarly, see next.
What will be the next case?
If I see the next case.
We will do the same way using the same proof.
Suppose we have next.
If we have a-1 and if we are asked to divide by a plus 1.
Similarly, you try once that how we can change it and do it.
Try once, it will be same as it was done.
And in that, it will be applied, last point.
So, how we will do this?
If we do plus one, plus one, minus one, because we want a
which is below to be up.
Now below is a plus one.
So, what should come up? a plus one.
So, plus one, minus one, a plus one, and minus one, whole
power n, divided by a plus one.
Now see, this term is one, this term is one, and we have k,
k becomes minus k.
See here, so that second property, remember this minus one.
So, what will happen?
Its remainder will be minus one to the power n.
So, what will be its answer?
Minus one to the power n, which will open two ways for us.
That is, its answer can be either minus one, when?
When we have n odd, or we can get its answer as one, when n
is even.
So, these were some points to find out your remainder.
More interesting topics will come in the future to find out
in unit digit.
We will use remainder in many places.
So, keep practicing till then.
Repeat all these divisions again and again, then only you
will be able to memorize and remember.
Thank you for watching.
That sounds interesting.
Count me in.
I'll make sure to spread the word to all my classmates.