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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