Hi everyone.
Am I audible to all of you? Please write
Okay, great.
Fine. So, welcome to this session by
Iwenta. Okay, the event CAD before CAT,
a very famous initiative that we do
every year. I'm just waiting for a
couple of more students to join and then
we'll get started. Okay, today's session
is based upon remainders and
Hi Gorav. Hi Anikit.
Good evening everyone. So let's get
started. U basically you might be
knowing the agenda by now. We are doing
these daily sessions in the event CAD
before CAT and it is actually CAD before
CAD because this entire initiative is an
intensive drill I would say drill rather
than just random classes. It's a very
good drill just s concepts cover. Okay.
And this is a 50-day initiative. You
might be knowing about it by now. Week 1
2 3 4 5 6 7 weeks event
session week.
I think you must have already gone
through factors or lecture right today.
I'm here to deal with divisibility and
remainder. And as you know CAD 2024 I
think remainder say question right? So
remainders is easy to learn and very scoring.
scoring.
I would suggest
as an aspirant
okay because a lot of students are going
to participate in this. In fact I guess mog
mog
iat 7 all India open mog a lot of
students will be writing it I think iat
so you'll be knowing right
right
and that could be a wakeup call for many
of you so do come to come to this
session every day I'm fat questions
and it's a good revision for for you
obviously. Okay.
So factors
LCM factorious factorious and
daily
issue, isn't it?
We have first question on your screen
and remainders favorite because
calculation mistakes students okay you
don't have to do any such calculation
mistake if you deal with calculations
and all question
and remainders question they are always
solvable you will always have a way to
solve remainders questions okay you can
see first question on your screen do
give Give it a try answer and then I'll
Okay.
Find the remainder when this power this something
something
is being divided by four. Okay.
remainder.
but it helps us to solve the questions
if you divide four by 7
definitely I'll continue in English okay
not an issue
negative remainders if you divide four
by 7. What will be the negative remainder?
minus 3. Correct? Okay.
So, negative remainders usually exist
when our divisor it is I would say
greater 7 is your divisor, four is your
dividend, isn't it? So, suppose you
divide four by 7. What will you do? If
you are finding out positive remainder
in positive remainder what you do 7
zeros are zero you get four this four is
what positive remainder but if I tell
you to find out the negative remainder
you divide 4 by 7 7 1 are 7 4 - 7 you
get minus 3 this is your negative
remainder isn't it?
So just take care of negative remainders
it helps us to
ease our calculations isn't it? Now let
me solve this question. I have 4 ^ 41 /
+ 7 ^ 41 + 11 ^ 41
+ 17 ^ 41
oh by 7 by 4 okay this is being divided by
by four
in remainders We have a couple of
theorems as well. What are those
theorems? I'm not using them here. I'm
just telling you, okay? Because we'll be
using them in a while. Euler Euler
theorem is used in case of high powers,
isn't it? When we have high powers,
then we have another remainder called
Chinese or we in short call it CRT.
Chinese remainder theorem. It is used
when uler fails. What we do in CrT? We
break the divisor right
right
break the divisor into
into
product of two co-primes
product of two co-prime numbers right
this is when we have crd and finally
when we deal with factorials we have
Wilson remainder theorem here if you
solve this question I know that my
divisor can disturb the basis Is isn't it?
it?
Powers here are 41 41 41 41 41 bases are
4 7 11 17 25 from the divisor you can
directly divide the base and replace it
with remainder. So from four if you
divide 4^ 41 obviously when this four
divides this base 4 what do you get the
remainder 0 and 0 power 41 gives you
zero only here 7 divides uh I mean four
divides 7 if four divides 7 what will be
the negative remainder positive
remainder will be three isn't it if 7 is
divided by 4 you get three as the
remainder but if you divide 7 by 4
negative remainder will be minus 1 so -1
then again if four divides 11 if four
divides 11 what is the negative
remainder again 4 3's are 12 that means
minus one again I just explained the
concept of negative remainders right
then four divides 17 4 are 16 we get
positive 1 as the remainder
and finally 4 divides 25 you get you get
what 1 and 1 to the^ ^ 41 this entire
number will again be divided by 4 yes or no
no
so - 1 ^ 41 is I'll just solve it here -
1 - 1 ^ 41 is -1 and then there we have
Now finally you just have to these all
get cancelled. We finally get a 0
divided by 4. That simply gives you 0. Right?
H Aish man you can use that method as
well. A power n plus b power n divided
by a plus b when n is odd. Right?
Yes. Only cat. You're saying that sir
scratch I just explained from the very
scratch. If you go back and check the
negative the negative remainder part I
have used negative remainder only over
here. Okay. I have not used these
theorems. I just said that we in
remainders generally have uler remainder
theorem, Chinese remainder theorem and
Wilson theorem. Euler is used when there
is high power. Chinese remainder theorem
is used when we have high power but uler
fails. And we have Wilson remainder
theorem when there's a factorial. Right?
Yes or no? When we have factorial
remainder questions. So
even though we have high power here but
why did I not use uler? Because without
using uler as well if I simply disturb
the bases. This is the sum that I have
to divide by four. Four can disturb the
base. I mean divisor can always disturb
the bases directly and replace the bases
with respective remainders. So four
divides 4 remainder is 0. I I got a 0. 0
to power 41 is 0. 4id 7.
See in maths we always have to ease our
calculations. 4id 7 we can have two
remainders. positive remainder 3 and
negative remainder minus 1. So I took
minus1. Why? Because -1^ 41 is a smaller
number minus 1. Right? Then 11 is
divided by 4. What do you get? -1 - 1^
41. Then 4 divides 17 we get remainder
1. 4 divides 25. We get remainder 1. 1^
41. 1^ 41 that I have over here. So 0 -
1 - 1 + 1 + 1 all gets cancelled. 0
divided 4 remainder is 0. This is an
easy question. Okay.
You cannot expect such question in CAT
and unless they they decide to give it.
But um I mean to just clear the concepts
that divisor can directly disturb the
basis that is why we have used this. I
hope this question is clear right
and you should actually try to solve
remainder question within 1 minute.
Okay. Such type of remainder questions
under 1 minute.
Now try to solve this this question.
Okay. I'll read it out for you. Find the
remainder obtained when n is divided by
6 factorial where n is 1 factorial + 2
factorial plus 3 factorial + 4 factorial
plus 5 factorial up till 89 factorial.
So 1 factorial
+ 2 factorial + 3 factorial
plus 4 factorial
+ 5 factorial + 6 factorial plus up till
up till up till
89 factorial. This is being divided with
Now see
we can clearly see that one factorial
when it when it is divided by 6
factorial it's a small number isn't it 1
factorial is 1. If you divide 1 by 6
factorial remainder will be 1. Right? If
you divide two factorial, two factorial
is 2. 2 divided by 6 factorial remainder
is two. 3 factorial is what? 6. 6
divided by 6 factorial. 6 factorial is a
big number. Okay. 720.
So 6 divided by 720 remainder is 6. 4
factorial is 24. 24 divided by 6
factorial is 720. Okay. 24 divided by
720 remainder is 24. 5 factorial is 120.
Okay. 120 divided by 720 remainder is
120. So I can see that 1 factorial 2
factorial 3 factorial 4 factorial 5
factorial will give me some remainder
isn't it? As soon as I come to 6
factorial 6 factorial divide 6 factorial
what is the remainder? 0 6 factorial
divide 7 factorial. So over here if I
try to write down 7 factorial divided by
6 factorial what will be the remainder?
7 factorial can be written as 7 into 6
factorial and divided by 6 factorial. It
cancels that means remainder given will
be zero. So six factorial onwards each
and every factorial that I'm going to
get till 89 factorial each factorial
will have six factorial included
7 factorial will have six factorial 8
factorial will have six factorial 9
factorial will have six factorial you
know till 89 factorial. So obviously 6
factorial onwards each factorial is
going to give remainder zero when
divided by 6 factorial. Now 1 factorial
is 1, 2 factorial is 2, 3 factorial is 6,
6,
4 factorial is 24, 5 factorial is 120.
Right? This is divided by 6 factorial. 6
factorial is 720.
Yes or no?
153 divided by 720
153 is my answer. So this is also an
easy question but I think a similar
question has appeared in a lot of mocks
earlier you know or type question
we just get confused in some way key how
can I calculate this big sum you know so
there you have to be a bit smarter fine
I think this is also clear it's an easy
yes or no this is easy okay
let's go Go to next question. Okay, next
question. 3 ^ What will be the remainder
when 3 ^ 114 + 4^ 96 + 8 ^ 41 is divided
by 7? So 3 power 114 + 4 ^ 96 + 8 ^ 41
divided by 7. As I said in remainders,
you always have to come up with smaller
smaller numbers, smaller smaller
remainders. 7 divides 8. It is very
clearly visible that 7 divides 8
remainder will be 1. So this 8 will be
replaced with one. 7 divides 3 remainder
is three only. There is no such
replacement. 7 divides four remainder is
four. No such replacement. But can I
turn this four into something which on
being divided by 7 gives me a smaller number.
number.
Now here you have to be smart again. You
know that 4 cube is 64. 4 cube is 64 and
64 when divided by 7. 7 9 6 1 remains.
Can I just transform this 4 ^ 96 into
4 ^ 96 can be transformed to what? 4 cub
^ 32 isn't it? That means 64 ^ 32.
Yes or no? So in the next step can I do
3^ 114 + 64 ^ 32
+ 8 ^ 41 divided by 7.
I have I have dealt with two numbers.
Okay. 64 on divided by 7 gives remainder
1. 8 on being divided by 7 gives
remainder 1. I get 1 + 1 that means 2.
So now I'm solving over here after this
step. Okay. I still have 3 ^ 114 here
plus 1 + 1 / 7. Now this is something I
have to deal with. Can I transform three
into some other number? Three can be
made 3 square. It can be made 3 cube. It
can be made 3 ^ 4. Something like that.
If I turn this 3 ^ 114 into what? 27.
So 3 ^ 114 can be written as 3 cube to
the power
38. Isn't it? And 3 cube is what? 27. 27
^ 38
to 27 ^ 38 + 2 1 + 1 2 divided by 7. Now
this 7 can directly disturb this 27.
Okay. And 7 when disturbs 27 7 divides
27 I get negative remainder again.
Right? That is minus 1. So I get - 1 ^
38 + 2 / 7 - 1 ^ 38 is 1 1 + 2 / 7 1 + 2
that gives you 3. Right?
Right?
So all of you are experts reminded us already.
already.
Can we take four as minus 3 add 3^ 114
and 3^ 96 to get 3^ 18? No. Uh hurry
Krishnan. So you have done a serious
mistake over here. You are saying that
3^ 114 3^ 114 plus minus 3^ 96 should
become what? 3^ 18. No
powers cannot be you cannot do
subtraction of powers like that. Okay,
just recheck what you have done. Okay,
write it down on a piece of paper.
You'll find your mistake. I want you to
I hope that is clear or other way to
solve this question is you have a big
power here. Whenever we have a big power
then we should think of uler. What does
uler do? Uler reduces the power, isn't
it? But
when do you apply uler? You only apply
uler when the divisor and dividend are
co-prime. So three and seven are
co-prime. Yes. You know this I have done
without using any theorem. Okay. Because
transformation was easy for me. Right
here if you apply uler theorem in this
part right. So uler of 7 becomes 6. 6
divides power 114 we get a reduced
remainder which is two. So 9 9id 7 that
is I mean again you're coming up with
that only isn't it
yes or no why only 27 why not 81 because
if you turn this into 81 okay if you try
to turn this into 81 so what what will
you do you'll do 3^ 4 and then you again
need to distribute the power okay four
should divide 114 does four divide 114 no
no
you're stuck again I I mean you'll have
to do something like this uh
this 3^ 112 into 3² and then you again
divide by what 7. But 7 when it divides
81 does it give me any small number? No.
But when I did when I transformed this
into 3 cube 3 cube becomes 27 27 1 being
divided by 7 gives me a small number.
Try to generate small numbers in
There are thousand ways to solve one
single remainder question. This question
could have been solved with maybe
transformation is one way. You could
have directly applied uler as well.
Isn't it? Nobody's stopping you. Use
your own method in remainders. The most
beautiful thing is use any method. If
you are mathematically doing the
calculations correctly, then you will
never get your answer wrong. Okay?
I hope this is clear.
How does 3 by 7 become 3? 3 divided by
7. This is not a fraction. Okay.
Okay. In remainders class, whenever I'll
be doing this, that means I'm dividing
the number above by the number below. Okay.
So, Ria 3x 7 is not a fraction. Okay.
What was I doing? I was dividing this by
7. So, I'm still dividing 3 by 7 getting
remainder three. Okay,
I hope this is clear. In fact, this
should be clear because I have not used
uler even if you don't know uler to be
because I have not used any theorem.
I've used simple mathematics and the
concept of negative remainders that I
told in the very first I mean 2 minutes
right now you have another question
which is 19 factorial divided by 23.
Most of the people they know Wilson
theorem. Okay, this is a question which
will be solved by Wilson theorem. Why?
Because here we have a question 19
factorial divided by 23. So a factorial
divided by another number. This is a
question of Wilson theorem. My first
thought will be does 19 factorial have
23? If 19 factorial has 23, I'll
straight away cancel it and get
remainder zero. Right? But 19 factorial
does not have 23. That means it cannot
be solved just like that. What is the
other way around? Other way around is
Wilson theorem. Okay,
Okay,
Wilson theorem
is a is a remainder theorem which works
specifically in cases of factorial
remainders. But for Wilson also we have
some conditions. What what are those
conditions? Wilson has some proven uh I
would say results already for factorial
remainders. What are those results? First
First
if you have
u a number okay if you have any
factorial number okay like this p min -
2 factorial
I will write mod now okay mod p okay
okay let me tell you already that
p is a prime number okay so when does
wilson theorem actually work when
suppose there's a prime number and it is
dividing a factorial value which is two
less than that prime. Let me give you an
example 11 factorial
mod 13 isn't it?
11 factorial 13 is a prime number 13 is
dividing a factorial value which is two
less than 13. So this is one of the
proven results of uler theorem. Whenever
you will see such a such a situation the
answer is always one. Okay this is
proven. Fine. Second result is what?
If you are dividing factorial of a
prime number one less than the one less
than the prime. So you are dividing the
factorial of one less than the prime
number. Prime number is p. You are
dividing factorial value one less than
that of the prime. In such case
remainder is always p minus one. So if I
transform this question to something
like this 12 factorial mod 13 12
factorial divided by 13 remainder will
be 13 - 1 12
third proven result of Wilson is p min -
3 factorial
divided by the prime remainder is
p - 1 by 2
p - 1 by
Is this clear to everyone? These are the
proven results of Wilson theorem, isn't it?
Okay. Fine. Now let me work upon my question.
question.
Okay. One more thing. Wilson theorem
does not have any proven such result for
P minus 4 factorial divided by P. I mean
Wilson theorem. These are the three
results that you can use further to
solve your questions. But only these
three are proven. Okay. So P min - 4 factorial
factorial
divided by P. What will be the remainder?
remainder?
That is what I have over here. This is
my situation, isn't it? This is a prime
number. It is dividing a factorial value
19 factorial
mod 23.
19 factorial mod 23. Do I really know
the direct formula for this? No. But
what do I know that you have to make use
of in Wilson? I know that in according
to Wilson P - 3 factorial mod P is P - 1
by 2. So can I say that
20 factorial
I'll write over here. Okay, we know that
this is something that we already know.
We know that 20 factorial mod 23 will be
what? P - 3 factorial divided by P. What
is the answer? P - 1 by 2, isn't it? So
P - 1 by 2 22x2 that means 11. So that means
means
this is something that I know if I take
P minus 3 factorial mod B that is equal
to 1 according to
11 not 1. Okay according to Wilson right
yes or no take I know this but what do I
actually want? I want to generate 19
factorial mod 23. Okay,
you all know that this is my dividend,
this is my divisor, this is my
remainder, right? Yes or no? So can I
say that okay just imagine 20 factorial
is a huge number isn't it? That is being
divided by 23 the divisor and remainder
is this. So can I write according to
this equation I hope you all know this equation
equation
dividend is equal to divisor
divisor
into quotient plus remainder right you
all know this equation what is the
dividend here
what is the dividend 20 factorial right
to 20 factorial will be written as can I
write can I write 20 factorial as
dividend is equal to divisor what is the
divisor 23 Do you know the quotient? No.
Number 20 factorial when divided by 23
quotient will be huge. So I don't know
its value. Let me take the quotient as x
plus but I know the remainder it is 11.
So can I write down this right hand side
Can I write down this right hand side as
I have explained this is the dividend
divisor. It's a huge number being
divided by 23 quotient will be huge. I
don't know it. So divi dividend is equal
to divisor 23 into quotient sum x plus
remainder. Now what do you want? You
want 19 factorial mod 23. This is what
you need to generate. How can I write
down 20 factorial? 20 factorial can be
written as 20 into 19 factorial, right?
Mod 23 and that should be equal to 23x + 11.
11. Right?
Right?
You have almost got what you needed. But
when will you get this value? When this
20 goes below this number, isn't it? So
this 20
19 factorial
mod 23
Has everybody understood this till this
Okay, good.
Your entire
answer lies here. How? You just need to
solve this. How will you be able to come
out with your answer? You just need to
manage a number above which is divisible
by 20. Question. This was my situation.
I took the closest situation to it which
was P min - 3 factorial mod P which can
be written as remainder will be 11.
Right? In this case I wrote down this as
23x + 11. Right? Now 20 factorial was
broken as 20 into 19 factorial. This now
I need this. This will be equal to 23x +
11 by 20. You just need to manage a
number above which is divisible by 20.
So if you put
if you put one over here x= 1 what will
you get? You'll get 34 divided by 20. Is
34 divisible by 20? No. You put x equal
to 2 you get 46 + 11. 57 divided by 20.
Is 57 divisible by 20? No.
You put x equal to 3 over here. What do
you get? You get 69 + 11 80. 80 divided
by 20. Is 80 divided by 20? Yes. So at
at x = 3 when I put x= 3 over here in
this I get 80 by 20 that gives me
remainder four and that is my answer.
So essay questions is go extension of
Wilson theorem extension I mean you knew
this already but you have an extension
some way you have to extend this right
so in such questions suppose 18
factorial your homework is solve with 18
factorial over here okay try to number
isn't it
you'll break it down again you'll start
with this and then you will take 20 into
19 over here 18 factorial mod 23 and try
to solve the rest of the Okay,
Okay. Uh how come three? Because three
is the only value at which this is okay.
Okay. Tell me one thing. What is this
over here? This is remainder.
This is remainder, isn't it? 19
factorial mod 23 means I need remainder
value. when this 19 factorial will be
divided by 23 to remainder will be an
integer positive number you remainder
yes or no.
So this remainder has to be equal to
what 23x uh plus 11 by 20 for this to
become an integer positive integer. I
need to put such value of x. I will
always start with 1 0 1 2 3 and so on so
on. X equal to0 this is not divisible by
20. X equal to 1 you get 34 divided by
20. 34 is not divisible by 20. X equal
to 2 you get 57 57 divided by 20 no 57
is not div by 20 x= 3 it becomes 69 69 +
11 is 80 80 divided by 20 you get four
that's your answer that is when I'm able
to generate a positive integer right
right
so 20 factorial is dividend why did we
write 20 factorial mod 23 as dividend
23 divisor Right?
Right?
Yes or no? This is my divisor. This is
my dividend. Okay?
Just go back and listen to what I said
again. Okay? We have few more questions
to solve. Okay? Four is the answer, right?
right?
You just need to watch this video once
again and you'll get it. Fine. Difficult.
Difficult.
In fact, you can take some questions for
yourself. Change the new values. is I'll
just take take a homework practice 37 37
is a prime and four reduce 33 33
factorial divided by 37
try okay to result already 34 factorial
mod 37 is equal to what already
18 and then you do the math okay that's
your homework
now we have another question to teach us
a very good a very good I would
uh technique as well and uh something
which can get you stuck in the exam. Say
find the remainder when 20 factorial is
divided by
289. There are two ways to do this question.
question. Okay.
Okay.
What do you see? 20 factorial divided by
289. 289 289 is 17² isn't it?
And 20 factorial is 20 into 19 into 18
17 into 16 factorial something like this.
this.
Can I write it like this? 20 factorial
17 squared. I can see that above I have
117 below I have a 17 I mean two 17s.
117 cancels 117 in the denominator. 117
has been cancelled. But you cannot
cancel numbers like this. Whenever
you're solving remainder questions, do
not cancel like this. Okay? Why? Because
if you are cancelelling the something
from the divisor, you are changing the
divisor which is not allowed. If you
simply cancel 17 and forget it, you have
changed the divisor. You cannot do that.
So the 17 that you have cancelled keep
it over here very very safe because it
is going to finally define my answer. Okay.
Okay.
What is going to happen here? I get 20
19 18
17 is gone. I'm left with 16 factorial
divided by 17. Yes or no? The question
looks very much doable now because 17
will divide 20 remainder 3. 17 divides
19 remainder 2 17 divides 18 remainder 1
and this becomes what 16 factorial
divided by 17 17 is a prime number 16
factorial is what one prime I mean prime
value which is one less than 17 right
we'll use Wilson theorem so if I ask you
guys 16 factorial
mod 17 what will be the answer
this is the perfect case of what p - 1
factorial mod p
such case remainder P - 1 isn't it? So P
- 1 that means 17 - 1 which gives you
16. So this particular part is going to
17 divides 20 remainder 3. 17 divides 19
remainder 2. 17 divides 18 remainder 1.
And 17 divides 16 remainder is
16. Isn't it?
And further 17 17 cancel it is not
getting cancelled. Right? Now what do
you have? You have 16 divided by 17.
Yes or no? 16 divided by 17 is -1. So
this is -6 divided by 17. Yes or no?
How? 17ide 16 remainder is -1. 3 into 2
into - 1 is -6 divided by 17. Now -6
divided by 17 is what? It is 11. Say -6
/ 17. How is the remainder 11?
You take minus common and deal with 6
divided by 17 inside. When you divide 6
by 17, you get negative remainder of
what? -1, isn't it? So minus is outside.
I get -1 divided by 17. And that simply
means minus minus I get + 11. So 11
Okay.
17 / 6 you get - 11 - 11 into minus is +
11. So 11 is the remainder isn't it?
This entire problem when I solved it I
am getting my final remainder as 11. But
that is not the answer. That is not the
answer. In fact, they will definitely
plant 11 as one of the options. They
should have done it. But this is your
final remainder. Your number already
cancel the 17. You already cancelled 17
when we started, isn't it? You will have
to multiply this 17 back with the
remainder you have got. And that gives
you 287
as your
187 as your final answer. Yes or no?
Which is option A. So this is also a
very good question in which we broke
down 20 factorial and 289
because they
isn't it? 20 factorial 289 divide. What
did we do? We broke it down. We
cancelled 17. We kept it safely over
here. Then we went on with rest of our
problem. We got final remainder 11. and
11 will finally be multiplied with what
was cancelled before and that got me 11
into 17 that is 187.
Rakkesh Kumar. So Rakkesh Kumar you want
a more theoretical and detailed video on
this. I suggest you to just go to
YouTube and search for um Chinese and
uler remainder theorem. I have made a
video. It's uh I would say it's a video
in which I have clearly explained uler
Chinese Wilson all remainder theorems.
It's a detailed video. You shouldn't you
should go and watch it. Okay. In that I
have explained the concept as well. Fine.
In fact in this one as well I explained
Wilson theorem isn't it in the previous
slide and we're just using the concept.
I I have not even used any concept over
here. Right.
Yes. True in all cases.
Your name is Sai Sai Bharda. Okay. So
Sai whenever you cancel any number
during your calculation you have to
But cancelled seven was in denominator.
an you need more practice. Okay, we'll
we just transformed this, cancelled the
17, kept it safely, kept going on with
our problem. We got remainder 11. But I
told you you cannot simply cancel the
divisor because if you simply cancel the
divisor, then you are changing the
divisor which is not allowed. Question
right
okay some of you are asking
why did you multiply by 17 okay you have
come till this point isn't it let me use
another pen you have come till this
point - 6 divided by 17 you got
remainder 11 divided by 17 but was 17
your actual uh divisor no your actual
divisor was 289. So you again have to
finally make this 17 289.
You have to multiply the numerator and
divi denominator with 17. Isn't it? 11
into 17 287 divided by 289. Original
divisor original divisor 289. You are
getting 11 into 17 287 as the answer. Okay.
Clear?
Fine. So there are multiple ways to
understand this. This was a
multi-conceptual question. I mean they
have not straight away given you the
question. You just have to break it
down, transform it. Let's go to the next
one which is 33 ^ 199.
33 ^ 199
into 26 factorial divided by 29.
This you can do isn't it? 29 divides 26
factorial. 29 is a prime. It is divid
dividing a factorial value which is
three less P minus 3 factorial mod P
- 3 factorial mod P according to Wilson
theorem is P - 1 by 2. So what is your
P? P is 29. Your prime is 29. This is
only applicable for prime numbers. Okay.
So 29 - 1x2 that is 28x2 that is 14.
That means this part is going to give me
what? 14. Let me keep the remainder. 14
here. This part. Now 26 factorial will
disappear. Okay.
33 power 199 divided by 29. Okay.
Okay.
I hope you clear
Yes sir. But 23 fac 20 factorial equal
to 23 x + 11 by this. we replace this
this this this I did not understand it
Koshi agraal I'll go back okay once we
reach the end of the session do remind
me I'll go back and explain that okay
let's try to solve this question first p
min - 3 factorial divided by p that
gives you 14 so I replace this part with
14 now 29 can directly divide 33 bases
can be directly disturbed by the divisor
so 23 disturbs 33 29 disturbs 33
remainder will be 4 ^ 199
into 14 divided by 29.
Now just before applying any sort of
theorem try to see whether four can be
transformed. This 4^ 199 be transformed
into something which on being divided by
29 gives you a smaller number.
4 square 16 16 divided by 29 does not
give you any small number. 4 cube uh 64
64 on being divided by 29. Does it give
you any smaller number? No. Not like 1 0
2 not nothing nothing smaller. Right
now this is a case in which
transformation is not that easily
possible. In such cases when you have
huge powers transformation is not
possible. Go to uler.
Why uler? I told you uler helps you
what? in high powers.
And why do we use uler in high powers?
Because uler reduces the power
reduces the power. Right?
Right?
Uler theorem reduces the power. But for
uler divisor and dividend should be
co-prime. So let me keep this 14 aside
for a while. Okay. Let me keep this 14
aside for a while. Let me just deal with
4^ 199 divided by 29.
Okay.
4^ 4 4^ 199 divided by 29. I'm dealing
with this huge power. 4 and 29. 4 and 29
are co-prime, isn't it? Yes. They have
HCF1. When are two numbers co-prime?
When they have HCF1. So 4 and 29 are
co-prime. We have a huge power question
as well. We can apply uler. How to apply
uler. In applying uler, first step is
checking whether the division divisor
and dividend are co-prime. Yes, they are
co-prime. If they are co-prime, next
step is finding out uler value of the divisor.
So, we have to find out uler value of 29.
29.
How many of you know how to find out the
uler value of any number?
Those who do not know, write no. Those
How many of you do not know how to find
No sa you don't know prair you don't
know diia you don't know I'm happy to
see so many nos here because at least
you guys will now know value okay if
probably you would have attended iiconda
classes they would have definitely
taught you I class I think
who does not know how to find out the
value or who is hearing the term uler
probably you can go for the last batch.
Okay, we are going to teach this in very
very very detail over there session
value. Let's talk about uler value. I
I'll just cancel this. Okay.
Let me cancel this. Let me teach you how
to find out the uler value. Okay. Euler
value. Suppose I ask you value of value.
Okay. We can find out uler value of any
number. Suppose I'm finding out uler
value of 10. Euler value of 10. You're
finding out ul value of 10. That means
first you do the prime factorization of
10. 10 is 2 into 5. Okay. The uler value
of 10 will be 10 into 1 - 1 upon first
Close the bracket. 1 - 1 upon 2 prime
which is 5.
So 10 into 1 - 1x 2 is half 1 - 1x 5 is
4x 5 2 into 5 10 is cancelled we get
four four is the uler value of 10. So if
I write the formula of uler value of any
number n it will be write down the
number first then 1 - 1 upon say a where
where a is what the first prime okay
into 1 - 1 by b which is second prime
and so on so on so on 1 - 1 by c 1 - 1
by d whatever is the number of primes.
Suppose I ask you another question.
I I'll I'll erase this as well. Okay?
Because we ultimately have to deal with
Suppose let us find out the uler value of
of
100. Okay. U value of 100. What to do?
First write down the prime factorization
of 100. 100 can be written as 2² into 5
square. Right? There are two primes.
There are two prime factors of 100. 2
and 5. So what will be the uler value of
100? 100 into 1 - 1 upon first prime
which is 2 into 1 - 1 upon 2 prime which
is 5. That gives you 100 into 1x 2 into
4x 5. 2 into 5. This cancels 10 into
Now we are dealing with what? Euler
value of 29. 29 prime factor. Can anyone
tell me how many prime factors does 29 have?
Okay, Somi, you're saying two
zero. RI is saying zero. 29 a prime
factor here 29.
29 has two factors 1 and 29 because 29
is a prime number. One is neither prime
nor composite one prime account. Okay.
29 has only one prime factor. So 29 the
number itself that means 29 into 1 - 1
upon first prime factor which is 29 into
1 - 1 upon second prime. But do we have
second prime number prime factor for 29? No.
No.
29 into 28 by 29 that cancels you get 28
or if I tell you a shortcut if I tell
you a shortcut uh
uh
let me write it over here uler of any
prime number
is always
that prime number minus one
okay just remember uler of any prime
number it's a shortcut it is always
prime minus one so suppose Uler of seven
it will be six. Euler of 13 it will be
12. Euler of 29 it will be 28. What does
this uler value do? This uler value
reduces the power.
Reduces the power.
What was the power I was dealing with? I
was dealing with this huge power 4 power
199 power. It's 199 power. How is this
uler value going to reduce it? the uler
value divides the power
and we replace the power with the
respective remainder. So when 28 divides
When 28 divides 199 remainder is three.
I'm buddy power 199 we will replace with
the respective remainder that means three.
three.
Yes or no? Y 4^ 199 it has been reduced
to 4 cube. 4 cube divided by 29 and 4
cube is 64. 64 divided by 29 is a it's
my 29 because this was a uler situation
right 29 finally reduced the power 199
to three the question has transformed into
okay huh value powers
We just need to take care of how many
prime factors does the number have. So
number into 1 - 1 upon first prime,
second prime, third prime, whatever is
the number of prime factors. Okay.
Now let me finally solve my question
This thing has been reduced to 4 cube a
14 and divided by 29. 4 cube is 64. 64
divid by 29 remainder. 8.
4 cube is 64. 64 divided by 29.
Remainder is
58. 6 I guess. Six. Sorry. Right. Six.
84 / 29. You have reached the end.
Answer is
26. 29 58. 26.
That is the answer.
Okay. Chinese
Clear? So do follow this Chinese again
I'm saying just search icon uler and
Chinese remainder theorem on YouTube
video it's thumbnail uler and Chinese
remainder theorem click it 15 minutes
video you'll always be clear with
now we come to questions where logic is
involved if the last two digits of 1
factorial 2 factorial 3 factor factorial
4 factorial plus plus till 119 factorial
is ab we have to find out their last two
digits then the value of a into b a into
b is to y deal you have to deal with the
last two digits it's question 1
factorial is 1 2 factorial is 2 3
factorial is 6 4 factorial is 24 okay
okay
5 factorial onwards
5 factorial you'll get 120. Isn't it?
When you do 5 factorial you get 120.
to 1 factorial 2 factorial 3 factorial 4
Let us think one thing
last two digits from 10 factorial
onwards. If you think about 10 factorial
10 factorial
if you talk about 10 factorial 10 factorial
factorial
you'll start with 10 10 9 8 7 6 5 4 5
4 3 2 1. So that means in the product 10
factorial the entire product we'll have
1 zero from here and 1 0 is in 10. That
means whatever is the product 10
factorial onwards 10 factorial onwards
each and every number
till 119 factorial will have two or more
zeros. Yes or no? 10 factorial onwards
all the numbers will have two or more
zeros. When you have last two digits as
zeros, do they contribute to the sum
1,000 here or 100 here or 25 here or 22
in sum suppose I'm just taking a
situation example okay 1,00 125 22 if I
ask you to add these numbers 5 + 2 7 2 +
2 4 did these two zeros from th00and or
these two zeros from 100 have any impact
on the sum of last two digits? No. That
means whenever we have any any such
number which has last two digits as
zeros, why I'm talking about last two
digits every time because last two
digits is written. I'm adding the last
two digits of all these numbers. If I
have last two digits as zero, they do
not contribute to the sum. That means we
will remove all such numbers from this
series which have last two digits as
zeros and they will start coming 10
onwards 10 factorial onwards to
we only have to pay attention till 9
right last digit I'm saying zero so 1
factorial 2 factorial 3 factorial 4
factorial 5 factorial is 120 120 last
two digits are 20 6 factorial last two
digits are 720 20. We'll take 20. Last
two digits
factorial. 7 factorial.
720 into 7,
isn't it? 7 factorial is 6 factorial
into 7. 6 factorial into 7. So 720 into
7 0 or four. 40, isn't it? Last two
digits. So 40, right?
right?
So we have reached till 5 factorial 6
factorial 7 factorial 7 factorial 8 factorial in 8 factorial
7 factorial 8 factorial in 8 factorial what will be the last two digits to 8
what will be the last two digits to 8 factorial 8 factorial is 8 into 7
factorial 8 factorial is 8 into 7 factorial and when it is when the entire
factorial and when it is when the entire thing um I mean the discussion is about
thing um I mean the discussion is about last two digits we only deal with last
last two digits we only deal with last two digits to seven factorial last two
two digits to seven factorial last two digits
digits 40
40 40 into
40 into 320 320 last two digits 20. That means 8
320 320 last two digits 20. That means 8 8 factorial last two digits 20.
Yes or no? And 9 factorial 20 into 9 8 factorial 8
And 9 factorial 20 into 9 8 factorial 8 factorial 9 mip you get 9 factorial to 9
factorial 9 mip you get 9 factorial to 9 factorial last two dates 20 into 9 that
factorial last two dates 20 into 9 that means 180 last two digits will be 80.
bus 1 factorial 2 factorial 3 factorial 4 5 6 7 8 9 factorial 10 factorial
4 5 6 7 8 9 factorial 10 factorial onwards s numbers last two digits will
onwards s numbers last two digits will be double zero they do not contribute to
be double zero they do not contribute to the sum of last two digits hence we
the sum of last two digits hence we don't care about them
80 20 100 100 100 key last two digits fit zero not
100 100 key last two digits fit zero not going to contribute to the sum Right.
going to contribute to the sum Right. Next we have 24 + 6 30 33 in
Next we have 24 + 6 30 33 in from here we get 80. From here
from here we get 80. From here what do I get? 113
what do I get? 113 113 last two digits sum last two digits.
113 last two digits sum last two digits. Finally add up that will give me 13 as
Finally add up that will give me 13 as the last two digits and that will be a 1
the last two digits and that will be a 1 13. We need the value of a into b
13. We need the value of a into b 1 into 3. That gives me three.
1 into 3. That gives me three. That's your answer. Three. Most of you
That's your answer. Three. Most of you are able to find out now, right? So
are able to find out now, right? So these are logical questions. Question.
these are logical questions. Question. Okay.
Basically calculation practice, brain practice. Okay.
practice. Okay. I hope this is clear.
Okay, let's try to solve this question. We just have two three more questions
We just have two three more questions left and then we'll get going. Okay,
remainder when 3 + 3 square + 3 cube and so on so on. This question a lot of
so on so on. This question a lot of people face difficulty in understanding
people face difficulty in understanding this question. Why? Because find the
this question. Why? Because find the remainder when 3 + 3 square + 3 cube
remainder when 3 + 3 square + 3 cube plus plus+ till 100 + 100 square + 100
plus plus+ till 100 + 100 square + 100 cq. Obviously 3 + 3 square + 3 cube.
cq. Obviously 3 + 3 square + 3 cube. Next 4 + 4 square + 4 cube. Then 5 + 5
Next 4 + 4 square + 4 cube. Then 5 + 5 square + 5 cube. 6 + 6 square + 6 cube.
square + 5 cube. 6 + 6 square + 6 cube. And so on so till 100 + 100 square + 100
And so on so till 100 + 100 square + 100 cube. Obviously after this there'll be a
cube. Obviously after this there'll be a term with 4 + 4 2 + 4 cq then
term with 4 + 4 2 + 4 cq then 5 + 5² + 5 cq and so on so on so on till
5 + 5² + 5 cq and so on so on so on till this point right. So 3 + 4 + 5 till 100.
this point right. So 3 + 4 + 5 till 100. So 3 4 5 until 100. This will be the sum
So 3 4 5 until 100. This will be the sum of one of the brackets.
of one of the brackets. Similarly, I'll have 3 square + 4 square
Similarly, I'll have 3 square + 4 square + 5 square till 100 square
right sum of all squares. And similarly I'll have 3q + 4q + 5q till 100 cq.
Isn't it right? What is this? Sum of first n
right? What is this? Sum of first n natural numbers. Something like sum of
natural numbers. Something like sum of first n natural numbers. This is like
first n natural numbers. This is like sum of squares of first 100 natural
sum of squares of first 100 natural numbers. This is like sum of cubes of
numbers. This is like sum of cubes of first
first 100 natural numbers. But in this what is
100 natural numbers. But in this what is missing? Three is missing. Three is
missing? Three is missing. Three is miss. I mean 1 + 2 is missing. Right? 1
miss. I mean 1 + 2 is missing. Right? 1 + 2 means three is missing.
+ 2 means three is missing. So from from the sum of first n natural
So from from the sum of first n natural numbers I just need to remove three. So
numbers I just need to remove three. So what is the sum of first n natural
what is the sum of first n natural numbers? It is n into sum of first n
numbers? It is n into sum of first n natural numbers is n into n + 1 by 2.
natural numbers is n into n + 1 by 2. Sum of squares of first n natural
Sum of squares of first n natural numbers is n into n + 1 into 2 n + 1 by
numbers is n into n + 1 into 2 n + 1 by 6. Sum of q of first n natural numbers
6. Sum of q of first n natural numbers is n into n + 1 by 2²
right n into n + 1 by 2 this will be the sum
n into n + 1 by 2 this will be the sum of 1 200 this will be the sum of 1 200
of 1 200 this will be the sum of 1 200 but I don't need
but I don't need but I need to remove 1 + 2 from this
but I need to remove 1 + 2 from this that means three will be gone right this
that means three will be gone right this will be the sum
will be the sum Okay, I'll finally divide it by 49 as
Okay, I'll finally divide it by 49 as well. Okay,
well. Okay, sum of squares of first 100 natural
sum of squares of first 100 natural numbers. N
numbers. N into n + 1 into 2 n + 1 divided by 6.
into n + 1 into 2 n + 1 divided by 6. From that I need to remove 1 square + 2
From that I need to remove 1 square + 2 square 1 square + 2 square is 5. Remove
square 1 square + 2 square is 5. Remove five.
five. Here sum of cubes of first 100 natural
Here sum of cubes of first 100 natural numbers n into n + 1x2 square to n 100
numbers n into n + 1x2 square to n 100 into n + 1 by 2²
we don't have 1 cube + 2 cube here to 2 cube is 8 1 cube is 1 that means 9 will
cube is 8 1 cube is 1 that means 9 will be removed and this entire thing will be
be removed and this entire thing will be divided by 49.
divided by 49. You many of you might be feeling key
You many of you might be feeling key this is very complex to solve but 2
this is very complex to solve but 2 cancels 50 right. Can six cancel 2011
cancels 50 right. Can six cancel 2011 they go 6 can be written as 3 into 2
they go 6 can be written as 3 into 2 right? 2 cancels 100 that means 50 and 3
right? 2 cancels 100 that means 50 and 3 cancels 201 that means 67 right? This
cancels 201 that means 67 right? This two cancels 50.
two cancels 50. Okay.
Okay. Yes or no?
50 into 101us 3. Can I replace this with 50 into 101 - 3?
1 - 3. Yeah. Yeah. here 50 into 101 into 67 - 5 50 101 67 So let me just remove
67 - 5 50 101 67 So let me just remove it
50 67 - 5
67 - 5 and then we'll just erase rest of the
and then we'll just erase rest of the part
This is what I get from here. Here I'm getting 50 squared 101 square - 9. to 50
getting 50 squared 101 square - 9. to 50 squared
squared 101 square
50 squared 101 square - 9 right
101 square - 9 right this is what I finally get and this
this is what I finally get and this entire thing has to be divided with 49
entire thing has to be divided with 49 9. Now the question is going to become
9. Now the question is going to become easier. Why?
easier. Why? Because if you divide this entire thing
Because if you divide this entire thing by 49, 49 divides 50 gives you remainder
by 49, 49 divides 50 gives you remainder 1. 49 divides 101 gives you remainder 3.
1. 49 divides 101 gives you remainder 3. 49 divides 50 gives you remainder 1. 49
49 divides 50 gives you remainder 1. 49 divides 10 1 gives you remainder 3. 49
divides 10 1 gives you remainder 3. 49 divides 67 gives you remainder 18. 49
divides 67 gives you remainder 18. 49 divides 50 into 50 to 50 divide 1. 50
divides 50 into 50 to 50 divide 1. 50 divide 1. 1 into 1. That means again
divide 1. 1 into 1. That means again remainder here will be 1. 1 divide 1 0 1
remainder here will be 1. 1 divide 1 0 1 that means 3. 3 square
that means 3. 3 square isn't it? What do I finally get? I
isn't it? What do I finally get? I finally get
finally get 0 because 3 - 3 is 0.
0 because 3 - 3 is 0. This is
54 - 5. That means 18 into 3 into 1 54 54 - 5 49 to 0 + 49
54 - 5 49 to 0 + 49 and you 9 9 - 9 0
and you 9 9 - 9 0 divided by 49 we finally get 49 divided
divided by 49 we finally get 49 divided by 49 that gives me remainder 0
by 49 that gives me remainder 0 is the answer
is the answer I hope this is clear
I hope this is clear fine
fine So you're again using the same concepts
So you're again using the same concepts little more. I mean I would say they
little more. I mean I would say they just tweaked the question. They just
just tweaked the question. They just want to see whether you take the correct
want to see whether you take the correct steps or not. If you take the correct
steps or not. If you take the correct steps I think you'll get the answer. I
steps I think you'll get the answer. I hope this is clear. We'll go to the next
hope this is clear. We'll go to the next one. Okay.
one. Okay. See these sessions are going to be
See these sessions are going to be recorded anyway. Okay. You can always
recorded anyway. Okay. You can always come back and check
this one. When 987654 3210 is divided by 102
98 76 54 32 1 0 divided by 102.
76 54 32 1 0 divided by 102. This is a problem in which you can
This is a problem in which you can either directly divide but 102 divide
either directly divide but 102 divide it's not easy right? It's not easy. Then
it's not easy right? It's not easy. Then another way is you'll try to use Chinese
another way is you'll try to use Chinese remainder theorem. What did I say? In
remainder theorem. What did I say? In Chinese diva theorem, you break down the
Chinese diva theorem, you break down the divisor into product of two co-primes.
divisor into product of two co-primes. 102 can be written as 6 into 17. But is
102 can be written as 6 into 17. But is it going to work?
it going to work? What do you do in Chinese?
What do you do in Chinese? In Chinese there is a divisor above. You
In Chinese there is a divisor above. You break the divisor as like this. I mean
break the divisor as like this. I mean 102 102 if you break into product of two
102 102 if you break into product of two co-primes. Those are 16 and 17, isn't
co-primes. Those are 16 and 17, isn't it? Uh 6 and 17.
it? Uh 6 and 17. We individually have to divide this by
We individually have to divide this by six, find out the remainder. If we
six, find out the remainder. If we individually have to divide this by 17,
individually have to divide this by 17, find out the remainder.
find out the remainder. But if you divide this by 17, it is
But if you divide this by 17, it is going to take a lot of time. You have to
going to take a lot of time. You have to do the long division method, right? Just
do the long division method, right? Just imagine you are dividing this by 17.
imagine you are dividing this by 17. There'll be a big long division problem,
There'll be a big long division problem, right? What is the shorter way out? We
right? What is the shorter way out? We see that here we have 102. Can I break
see that here we have 102. Can I break this break break this down like this?
this break break this down like this? This is a 1 2 3 4 5 6 7 8 9 10 digit
This is a 1 2 3 4 5 6 7 8 9 10 digit number. Can it be written as 98 into
number. Can it be written as 98 into 10 ^ How many zeros do we have? 1 2 3 4
10 ^ How many zeros do we have? 1 2 3 4 5 6 7 8. Can it be written like this? 98
5 6 7 8. Can it be written like this? 98 into 10^ 8 + 76 into after 76 we have 1
into 10^ 8 + 76 into after 76 we have 1 2 3 4 5 6. We need six zeros. 10^ 6 plus
2 3 4 5 6. We need six zeros. 10^ 6 plus 54 into 10 ^ 4.
Right? Then 32 into 10²
Then 32 into 10² + 10 this entire sum can be divided with
+ 10 this entire sum can be divided with 102.
102. Yes Aishwan you're correct. Okay. Now
Yes Aishwan you're correct. Okay. Now 10^ 8 is something it can be turned into
10^ 8 is something it can be turned into 100 and once it is turned into 100 102
100 and once it is turned into 100 102 will divide that 100 and give remainder
will divide that 100 and give remainder minus2. So 10^ 8 can be written as 100 ^
minus2. So 10^ 8 can be written as 100 ^ 4.
4. Yes or no? 10^ 6 can be written as 100 ^
Yes or no? 10^ 6 can be written as 100 ^ 3. 10^ 4 can be written as 100 ^2. And
3. 10^ 4 can be written as 100 ^2. And 10 square can be written as I mean if
10 square can be written as I mean if you divide 10 square by 102 100 by 102
you divide 10 square by 102 100 by 102 you get remainder minus 2. So let me
you get remainder minus 2. So let me replace it with respective remainders.
replace it with respective remainders. Okay.
Okay. This becomes 4 32.
This becomes 4 32. Okay
Okay 4 3 2. Now this 100 gets divided by 102
4 3 2. Now this 100 gets divided by 102 you get - 2^ 4 isn't it? This 102
you get - 2^ 4 isn't it? This 102 divides this 100 you get -2 cube right
divides this 100 you get -2 cube right this 102 divides 100 we get -2 square
this 102 divides 100 we get -2 square right
right so finally if I have to write this down
so finally if I have to write this down can I write it like this
98 into 16 98 into 16 76 into - Right.
Right. 54 into 4. 54 into 4 is 216.
54 into 4. 54 into 4 is 216. 216 plus minus 32 into -2 that means -
216 plus minus 32 into -2 that means - 64
- 64 + 10.
+ 10. Okay.
Okay. - 64 + 10.
Okay. Divided by 102
Divided by 102 is quite easier, isn't it?
is quite easier, isn't it? 102 again divides 98 and gives remainder
102 again divides 98 and gives remainder -4,
-4, right?
right? - 16 into -4. 16 into -4 is - 64.
- 16 into -4. 16 into -4 is - 64. - 64
- 64 then -76 into 8
- 216 sorry + 216 and this was + 64 over here. Okay. + 64 - 10 that means - 54
here. Okay. + 64 - 10 that means - 54 divided by again 102
divided by again 102 I guess solve right can you solve from
I guess solve right can you solve from this point onwards just find out your
this point onwards just find out your answer can you do it yes or no
I think it can be done right I have explained the approach how to break it
explained the approach how to break it down and why Chinese India theorem did
down and why Chinese India theorem did not work. Some other way did not work
not work. Some other way did not work and long division was a foolish idea
and long division was a foolish idea here. Right?
here. Right? Can you do it? We'll go to the next
Can you do it? We'll go to the next question and finish the session.
First term should be - 64. Yes, - 64. I have written minus only. Okay.
have written minus only. Okay. - 64. Okay.
Okay, let's let's go ahead. Okay, solve this question. Find out the answer. Uh,
this question. Find out the answer. Uh, when a natural number P is divided by
when a natural number P is divided by 18, remainder is four. So, P is divided
18, remainder is four. So, P is divided by 18. P is the dividend. Dividend is
by 18. P is the dividend. Dividend is equal to divisor. Divided is 18 into
equal to divisor. Divided is 18 into some quotient, they've not given you the
some quotient, they've not given you the quotient, so let it be X, remainder is
quotient, so let it be X, remainder is four.
four. Another statement is when a natural
Another statement is when a natural number Q is divided by 9, remainder is
number Q is divided by 9, remainder is 6, the remainder is R. when p + q is
6, the remainder is R. when p + q is divided by 3. Okay. When uh a natural
divided by 3. Okay. When uh a natural number q is divided by 9, remainder is
number q is divided by 9, remainder is six. So when it is divided by 9,
six. So when it is divided by 9, quotient will change, right? Number when
quotient will change, right? Number when we divide a number by two different
we divide a number by two different numbers, quotients are different, isn't
numbers, quotients are different, isn't it? So that is the reason I have taken
it? So that is the reason I have taken quotient y here and x here different
quotient y here and x here different quotients a remainder is six. The
quotients a remainder is six. The question is asking the remainder is r
question is asking the remainder is r when p + q is divided by 3. So let's do
when p + q is divided by 3. So let's do p + q. If you do p + q
p + q. If you do p + q that is 18 x + 9 y
that is 18 x + 9 y 18 x + 9 y + 10, isn't it? 18 x + 9 y +
18 x + 9 y + 10, isn't it? 18 x + 9 y + 10 divided by 3 that gives you what? 3
10 divided by 3 that gives you what? 3 divides 18x. Yes, 18 is a multiple of
divides 18x. Yes, 18 is a multiple of three or will always give remainder 0. 9
three or will always give remainder 0. 9 is divisible by 3 will always give
is divisible by 3 will always give remainder 0. So 10 divided by 3 that
remainder 0. So 10 divided by 3 that gives you remainder 1. What is this one?
gives you remainder 1. What is this one? This is the value of r because r is
This is the value of r because r is what? When p + q is divided by 3 what we
what? When p + q is divided by 3 what we are getting is r. So that simply means r
are getting is r. So that simply means r is 1. The question asks find the value
is 1. The question asks find the value of 16 - 4^ r 16 - 4 power r is 1 / 3 12
of 16 - 4^ r 16 - 4 power r is 1 / 3 12 / 3 answer is 4.
I hope this is clear. This is easy. Okay, this is not even I would say cat
Okay, this is not even I would say cat level. This is very easy.
level. This is very easy. Should we go to the next question?
Okay, good. Next. A number is formed by writing the first 500 natural numbers
writing the first 500 natural numbers next to each other as 1 2 3 4 5 6 7 8 9
next to each other as 1 2 3 4 5 6 7 8 9 10 11 12 up till 500. So this is a
10 11 12 up till 500. So this is a series of first 500 natural numbers.
series of first 500 natural numbers. Find the remainder when this number is
Find the remainder when this number is divided by 64. Easy question. Okay,
divided by 64. Easy question. Okay, those who have studied number system in
those who have studied number system in depth, easy question for them. What is
depth, easy question for them. What is your divisor? In remainder problems,
your divisor? In remainder problems, divisibility questions, a lot can be
divisibility questions, a lot can be answered through the number that we have
answered through the number that we have been given 64. It's a very special
been given 64. It's a very special number. What is special about 64? 64 can
number. What is special about 64? 64 can be written as 2^ 6. Right? You all know
be written as 2^ 6. Right? You all know that divisibility of any number of form
that divisibility of any number of form 2 power n. When you have two as the
2 power n. When you have two as the divisor, what do you check? When you
divisor, what do you check? When you want to check whether a number is delled
want to check whether a number is delled by two or not, what do you check? You
by two or not, what do you check? You check the last digit. Isn't it? Suppose
check the last digit. Isn't it? Suppose if a number has to be delled by two,
if a number has to be delled by two, what do you check? The unit digit. If it
what do you check? The unit digit. If it is divisible by two, number is divisible
is divisible by two, number is divisible by two. If I ask you the divisibility of
by two. If I ask you the divisibility of four, four is what? two square isn't it
four, four is what? two square isn't it for two you you check last digit
for two you you check last digit I mean I would say 2^ 1 last digit 2²
I mean I would say 2^ 1 last digit 2² that means you check last two digits
that means you check last two digits when you have to check the divisibility
when you have to check the divisibility of 8 is what 2 cube you check what last
of 8 is what 2 cube you check what last three digits right similarly if I go to
three digits right similarly if I go to 64 which is 2^ 6 what will to check last
64 which is 2^ 6 what will to check last six digits. So basically this question
six digits. So basically this question was planted to make you recall the
was planted to make you recall the divisibility rule of 2^ n. If there is
divisibility rule of 2^ n. If there is any number in the form 2 bar n its
any number in the form 2 bar n its divisibility rule is you have to check
divisibility rule is you have to check the last n digits to 2k^ 6 last digit
the last n digits to 2k^ 6 last digit check number last six digits.
check number last six digits. What are the last six digits of this
What are the last six digits of this number? This is a number which has been
number? This is a number which has been generated by writing down the first 500
generated by writing down the first 500 natural numbers next to each other. So
natural numbers next to each other. So last six digits will be 499
last six digits will be 499 50 0. That should be divisible by 64.
50 0. That should be divisible by 64. You find out the remainder and that will
You find out the remainder and that will be your answer.
be your answer. I I hope you can do this right.
49950 divided by 64. Long division,
Long division, you can find out the answer.
Yes or no? Last question for the day. Let's finish it quickly.
Find the sum of all three-digit even numbers that leave a remainder of four
numbers that leave a remainder of four when divided by five. If there is a
when divided by five. If there is a number which leaves remainder of four
number which leaves remainder of four when divided by five, what kind of
when divided by five, what kind of number that will be? Diviser is five.
number that will be? Diviser is five. Quotient can be anything. Rema remainder
Quotient can be anything. Rema remainder is four. So that number will be of the
is four. So that number will be of the form 5x + 4. But what do they want? They
form 5x + 4. But what do they want? They want threedigit even numbers. What is
want threedigit even numbers. What is the first threedigit even number that
the first threedigit even number that leaves remainder of four when divided by
leaves remainder of four when divided by 5? First three-digit number is 100. What
5? First three-digit number is 100. What is the first three-digit even number
is the first three-digit even number which leaves a remainder of four when
which leaves a remainder of four when divided by five? That is 104.
divided by five? That is 104. Next even number. Next even number of
Next even number. Next even number of three digits which when divided by five
three digits which when divided by five least remain of four is
least remain of four is 104 about
we'll just finish in 2 3 minutes. Okay. 114. Very good. 109.
Answer. Couple of you have answered 109.
Couple of you have answered 109. Somebody has even answered.
Somebody has even answered. Huh? 109 is wrong. It's It's odd.
It's odd. You need even to 114 is that next number. 124 is the next number.
next number. 124 is the next number. This simply becomes an AP. And what is
This simply becomes an AP. And what is the largest threedigit even number which
the largest threedigit even number which leaves a remainder of four when divided
leaves a remainder of four when divided by five? It is 994.
by five? It is 994. A
A sum terms
sum terms A 104
A 104 D is 10.
D is 10. It's an AP where first term and last
It's an AP where first term and last term are given. You can use the formula
term are given. You can use the formula n upon 2 a + l. That is one way. Another
n upon 2 a + l. That is one way. Another way is you use n upon 2 2 a + n - 1.
way is you use n upon 2 2 a + n - 1. Right? I hope you can do it. N upon two
Right? I hope you can do it. N upon two number of terms.
number of terms. Last term minus first term.
Last term minus first term. Last term minus first term divided by
Last term minus first term divided by what? Difference in 10 + 1
what? Difference in 10 + 1 890 890 by 10 18 + 1 90 90 terms series.
Okay. So sum n upon 2 a + l a is first term l is last term. So 90 by 2 first
term l is last term. So 90 by 2 first term a is 104
term a is 104 last term is 994. I hope you can find
last term is 994. I hope you can find out this sum
45 into 1098ish man 1098 into 45 yes you're correct. Okay,
into 45 yes you're correct. Okay, I hope the session was fruitful
Wilson or right and we solve some random questions that can be asked around
questions that can be asked around remainders. I think question I'll try to
remainders. I think question I'll try to bring them up in the next session. For
bring them up in the next session. For now that's it. I hope you enjoyed the
now that's it. I hope you enjoyed the session. Put down your feedback and if
session. Put down your feedback and if you need any improvement let me know.
you need any improvement let me know. And uh do solve numbers questions from
And uh do solve numbers questions from pyqs. Okay, numbers question p
pyqs. Okay, numbers question p because that actually tells you
because that actually tells you when you encounter a numbers question.
when you encounter a numbers question. Okay,
please one book which you can buy and study at home. You can book you can buy
study at home. You can book you can buy Iwanda books. Okay, material
Iwanda books. Okay, material there are a lot of books in the market
there are a lot of books in the market but
but it has been designed keeping in mind the
it has been designed keeping in mind the current pattern difficulty level of cat
current pattern difficulty level of cat okay that you can take and
less cluttered there are many books but they are cluttered
okay the exercises theory and solutions is all crap. Okay,
and solutions is all crap. Okay, better go for books.
They have enough questions, good solutions. Okay, all made by 99
solutions. Okay, all made by 99 percentylers.
percentylers. Okay, thank you so much. I'll see you in
Okay, thank you so much. I'll see you in few more days. I think I'll come up with
few more days. I think I'll come up with another session soon. Thank you so much
another session soon. Thank you so much for attending. Bye-bye.
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