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Divisibility & Remainders for CAT 2025 in One Shot | CAT Before CAT – Day 2

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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

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,

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

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

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

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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Divisibility & Remainders for CAT 2025 in One Shot | CAT Before CAT – Day 2