 So let's start real numbers and lots, you know, very important part of real number has been removed from your curriculum. So that's sad. And this is the final time I'm just revisiting this. So you know, where is the number system? Six marks. So you can expect six marks question from real numbers, right? But unfortunately, this has been removed. Euclid's division lemma is removed. But without that, this is the, you know, the most basic concept in your grade 10. But without that, I don't know. So the other topics which are left are which is which are there in this top and this chapter is fundamental. What is that fundamental law of arithmetic and theorem of arithmetic, sorry, and HCFL CM and then rational irrational numbers and things like that. Okay. So while picking up the questions I have picked only those where the Euclid's division lemma is not involved. So hence you will not be getting questions like prove that square of a prime number is of this form or sum of or product of three consecutive positive number is always divisible by six all these type of questions will not be there. Okay, so easy, easy topic it is very, you know, scoring you can get six marks very easily with very less effort. So we have done this so many times I'm not going to go back on to this. So there are six marks for this. So usually to one marker one, one, two marker and one three marker something one, sorry, one into one mark. So this could be the PS right now. So hence in this could be the configuration. So one into one mark. Okay, so this is one then one into two mark, maybe one into three mark, either this form or since the six marks or hence, two into one mark, and there is no four marker here. So, in the in your board exam so hence there will not be any four markers so either a five marker then which is unlikely. So one into five marker and one into one marker so this could be possibilities but in most likelihood I think this is going to be the scenario. Okay, or another one could be one into one and doing to do both. Anyway, so Euclid division lemma is very basic you know this. Though it is removed but still I have, you know, included this while in discussion because is the most basic how do how can you not study Euclid division lemma. So it says that given positive integers a and b there exists whole number q and are satisfying a is equal to Bq plus R which is the more sophisticated form of writing dividend is equal to that what you have seen dividend is equal to divisor. Into quotient. Plus the remainder. Right, so this is what is the basic thing and the remainder here is always less than the divisor it has to be. So that's what you already know and this is formulated in R. It was never told you that it is called Euclid division lemma until now. And very basic so no shows anyways. So next is, there is an algorithm to find out hcf. So the long division method of finding hcf you all know. Now, this is not there, but finding hcf lcm and all that might be there. So hence, you must know how to find out hcf. So you don't need to deploy Euclid division algorithm. Where a repeated division is done but let's say the question says that find the hcf of these two numbers and they will not say which algorithm or which process to adopt. Then you can always use this division algorithm though it is also I'm believing it's not there so those people who wrote the pre board in the month of December. Did you guys see any question related to Euclid division algorithm because the notification says only lemma is not there. So hence mostly division algorithm is also not there because it's directly linked to the lemma. So anyway, so we'll just move forward. Okay, so and the game starts with the fundamental theorem of arithmetic, which says that the fundamental theorem of arithmetic every composite number can be expressed as a product of primes. And this factorization is unique, except for the order in which the prime factors occur. So this is again very intuitive. We know that it's now formally, you know, introduced to you as a fundamental theorem of arithmetic, which is very, very true. So any composite number let's say 10 can always be expressed as two to the power one into five to the power one. So there are two aspects to it. One is this representation is unique. That means you cannot have any other prime factor apart from whatever we just found out by factorization. So that's unique and order could be anything. So you can write this as five into two to the power one also that's fine. That's not a problem, but you cannot have any introduction of any third. I thought algorithm was not there, but lemma was there. So this is what their official document says. Real numbers deleted portion nucleates division lemma do check it on CVC site. Okay, so this is the from there. Okay, so are you getting questions on lemma or in your way? Anyways, if you get also you know how to solve, but the formal syllabus starts from here, which is fundamental theorem of arithmetic and lots of application. Is it okay? So this is what it says. So for example, other example, let's take 24. So 24, you can factorize as four into six or eight into three. So that is two to the power three into three to the power one. Now this is again unique. So you cannot have any other any other representation like this. Right. So this is what the fundamental theorem of arithmetic says. Okay, enough. Uh, this is done. Every composite number can be uniquely except product of power. So primes in ascending or descending order. So in another way, you can write like that. So every composite number. So see it is talking about every composite number for that matter. Any number, any positive integer can be expressed uniquely as product of prime numbers including prime numbers themselves. So for example, you know, so you, you, if you don't count one. So hence 23 also can be written as, you know, 23 itself. So there is no product required here. So 23 is simply this, but yes, doesn't make sense for any prime number. Why? Because there is no product here. Okay. So hence we say for every other composite number. So 25 is you have to have five into five or five square. Okay. So this is unique. Okay. This is unique. Okay. Fair enough. We have, we will be having lots of application of FDA. Okay. So this is the first one. Let A be a positive integer and P be a prime number such that P divides a square and then hence P divides a. Okay. So what it means is if P divides so, and where are we going to use this? So you know that there will be one class of questions where you will have to prove that prove that root of five. He's a prime number. Oh, sorry, not a prime number. He's a, he's an irrational number. He's an irrational number. This is a question. It was asked last year also last to last year also every year one question typical. This question comes prove that root five is an irrational number and this particular theorem is used here. Which one? That is P. If A be a positive integer, let's say A is 25. Okay. So let A be a positive integer. So what is positive integer? Let's say A is, let A be a positive integer and P be a prime number such that P divides a square. So which one should I make? Let's say P divides a square. No. So let's say three and 36. Yeah. So let's say A is equal to 36. Okay. If A is equal to 36 and let's say P is equal to three. So you can see that P divides or other. Let's take A is six. Yeah. So that would make it easier. So P divides six square. Definitely P divides six square. Now when P divides six square, P also divides six. So that is what they're saying. Right. And you have to prove this. You have to prove proof. Can anyone suggest how to prove that that A be a positive integer and P be a prime number such that P divides a square. Then P divides A. So you'll use, use FTA here. Is it? What will we use FTA? How to, how to deploy FTA? So basically let us say A, let us say A. A can be expressed as, A is a composite number so it can be expressed as P1 to the power M1 into P2 to the power M2 into P3 to the power M3 into dot, dot, dot PN to the power MN. Right. So this is the expression for let's say the composite number by FTA. Right now. Fundamental theorem of arithmetic. This is what I can say. So what will be A square guys? So A square clearly will be P1 to M1. So every, everything will just get multiplied. Every, every M1 M2 MI will get multiplied by two. Right. And this is P3 to M3. So on and so forth PN to MN. Correct. So when I raise to a power of. Right. Right. Now if, if, so it is said that P divides A. P divides A square. That means what can I say P is one of. One of P1 P2 P3 dot, dot, dot PN. It has to be there is no other prime factor. So hence P has to be one of P is one of P1 P2 P3 PN because in the, in the factorization of a square we see only these N prime numbers are involved. So hence if P divides a square and P itself is a prime, that means P is one of P1 P2 P3. PN. Let us say P is equal to PK for some, for some K which is less than N. Okay. And it is always greater than equal to one. So either, so you could be P1 P2 P3 P4 P5 P10 whatever P some PK. Okay. Not the movie PK. So P subscript K. So far so good. Are you getting? And so it's PK is one of one of the prime numbers. Okay. Now clearly PK divides what P1 to the power two M one into P2 to the power two M two into so on and so forth. There will be a PK here. One of the prime numbers. And I'm writing at two and K into dot dot dot PN two M N. Isn't it? I can say that. Right. So PK is somewhere here, here. So there will be P. Yeah. So PK is this here somewhere. So clearly you can see that you can write this as P divides. So you can write this as P1 to the power M1 into P2 to the power M2 so on and so forth. And then here PN to the power MN and multiply it by the same thing P1 M1 P2. So there will be a PK here. Okay. Like that M2 and this is MK similarly PK here. So you can clearly see that PK is one of the factors. I'm not writing the full statement. So you can clearly see that PK. Right. PK is here also. And this is nothing but a. Okay. So clearly PK divides A or PK divides P divides A. That is what you need to prove P divides A. Right. So P was PK. So PK divides A. So PA divides. Yeah. So hence this particular thing is going to be used for proving what type of questions this one proves that route five is on your national number. You can always state this theorem to prove it. Okay. So there will be one question for sure on this type of this thing. There are infinitely many positive primes. This is another theorem to prove. Can you even try proving this infinite many prime and this has been there since centuries this particular theorem or concept that there are infinitely many positive primes since you know more than 2000 years ago. In fact, 2005 years ago people were aware that there are infinitely many positive primes. Okay. So how do we prove that? So again, very, very simple and again application of we will apply FDA here also application of FDA. What do we do? So let us say, you know, and the other technique which we're going to use is called contradiction method. So we will assume with let us say. Let us assume there are assume that there are finite number of primes finite number of primes. The moment you say that then obviously you'll have the largest prime number. And let us say Pm is the largest prime number largest. And because we are assuming that there are n prime numbers in total. So let us say Pn is the largest prime number. So there are n prime numbers without any doubt n prime number. So in subscript n, this subscript here, this means there are there are we are assuming there are n prime numbers only and it's a finite number. So in prime numbers, very good n prime numbers. Okay. So let us now design a new number. So let us say we are saying a is equal to P1 into P2 into P3 into dot dot dot Pn. We are designing a number which is product of all the first or all the n prime numbers. Okay. Now let us say Q is equal to a plus one. Okay. A plus one. Then what will this be? This will be P1 plus P2 plus sorry not plus into my bad. So P1 into P2 into dot dot dot into Pn plus one. So I'm just adding plus one to it. Okay. Now if you see clearly if P1 divides a then P1 will not divide a plus one. Right. So this if P1 divides a P1 will not divide a plus one right now. So two consecutive numbers are never divisible by same prime number. So hence P1 divides a P1 will not divide a plus one. So if P2 divides a which is given anyway I know then P2 cannot divide a plus one. Why? Because hence the lemma will again be used here because if you divide this a by anyone of P1 P2 P3 and always you will get a remainder zero. And if you're dividing Q by anyone of P1 P2 and all you'll always get a remainder one. And one is lesser than the smallest prime number. So hence these all will be true. So P3 divides a then P3 doesn't divide a plus one. Likewise, you can carry on this process and say if Pn divides a then Pn is not going to divide a plus one. That means none of none of P1 P2 dot dot Pn divides divides Q right. That means Q doesn't have prime factors prime factors. That means Q itself is a prime then Q is a prime. But Q is greater than P1 or you can write Q is greater than A which itself is greater than Pn right. Why it is A is greater than Pn? Because A was equal to P1 times P2 P3 times Pn all that. So hence A is definitely greater than Pn. So you're saying that means Q is a prime number. Prime number greater than greater than Pn greater than Pn. But how is this possible? Right. So you had assumed Pn to be the largest. That means this means Pn is not the largest or greatest prime number or this prime number. I mean that is so. So hence we need a contradiction. Hence it contradicts our initial assumption that there are finite prime numbers. Hence proved. All of you. So this is the proof for there are infinitely many positive prime numbers. So you design a new number which is product of all the prime numbers and add one, prove that none of the prime numbers divide that A plus 1 and hence there are infinitely many prime numbers. So far so good guys. Yes. All of you. Okay. Any questions so far? So maybe they can ask but I have not seen these kind of questions there in the board but good mental exercise. So this is an important result for two positive integers and A and B A cross B that is my product of A and B is always equal to XCF of AB and LCM of AB. Right. This is the one marker questions are sure short. So what there are several sets right in the board paper one set or a couple of numbers will definitely have one marker which has this particular concept in use. So what I'm saying is HCF into LCM is equal to AB so they will give you two numbers one number one LCM will be given find HCF vice versa like that. Okay. So this is we will see the use of this theorem later in the questions. Okay. Now another thing is the other type of questions will be P is a positive number prime then root P is an irrational number for example root 2 root 3 root 5 root 7 root 11 are all irrational numbers. That is what we are also going to see. And again, as I told you one question typically related to this is also there. We will see where we are meeting with the problem. Okay. Now this is another one marker types. So let X be a rational number whose decimal expansion terminates. Okay. Then X can be expressed in the form of P by Q where P and Q are co-prime and the prime factorization of Q is of the form 2 to the power m into 5 to the power n where m and n are non-negative integers. So what does it mean? So this is again very intuitive. We have seen this multiple number of times. So let us say let us take some decimal which terminates. So first is let's say 0.025. 0.25. This is a terminating decimal. So this is a terminating, terminating decimal. How do you represent this in form of P by Q? You have done this in ninth grade. So what is 0.25 as a P by Q form? What will you write 0.25 as in the P by Q form? What is 0.25? Are you there? Guys, 0.25 is how much? How much should I write? Anyone? 0.25 is how much? No one is responding. Guys, you're there. Hello folks. Yes. Slow response. Slow response. What is 0.25? Only two people are there. Everyone else is watching India Australia match. Is there some match going on? No. Is there any match going on? No match. Then where are people? Hello people. Can you respond please? 0.25 is how much? Or it is too lesser a question to be solved here. Anyways, so 0.25 is 1 upon 4. All of you know. And 1 upon 4 can be written as 1 upon 2 square. And this can be written as 1 upon 2 square into 5 to the power 0. So hence, any decimal which is terminating can be expressed like that. And another example could be let's say 0.3125. What is this? 3125? Can anyone tell me what is this? 0.3125. How do I convert it? How much is this? 1 upon something? How much is 3125? No one is participating. Hello people. Are you there? All right. There are 30 people here. They should be responding. Hello. What's the answer for this? 0.3125. Please convert it into B by Q form. So Sharguli is saying 25 by 8. How come? 25 by 8.3125. Sharguli. 5 upon 16. Do it and tell me all of you. Is it 5 upon 16? Is it 5 upon 16? Okay. So this should be very easy to calculate. 3125 pay 1 pay 114. Like that. Is it? Now it goes by 25 for sure. Yes. So how do you do this? This is 25. 1 then 62. So 24. 2 then 125. 5 divided by 25. 400. Okay. So again it is 5 upon 16. 5 upon 16. Right. Which can be written again as 5 upon 2 to the power 4 into 5 to the power 0. Isn't it? So this is going to be our terminating decimal. Right. Similarly, if I have let's say 1 upon 80. So 1 upon 80. Will this have decimal representation which is terminating or not terminating? Or not? Will this be having terminating decimal representation? Well, a quick 1 upon 80 terminating or not? Let's say a choose between these. So 1 upon let's say 640. C is 1 upon 120. D is 1 upon 375. Which one? Which one of these are terminating? A, B, C, D. Who? Which all? Terminating decimal. This is correct. A is correct. A is terminating. Yes. What about B? Is B terminating 1 upon 640? Is B terminating decimal representation? Come on, folks. Respond. All of you. Where is energy level? How will you get sent them? Hello, guys. Yes or no? B is terminating or not? Only 3 people. Shreyas, Arjun, Sargoli, Devjit. Okay. So Aryan has done all. Yes. This is terminating. This is not terminating because there is a 3 here. This is also not terminating. Okay. Why? So this is what we are learning. So hence if the denominator of the form 2 to the power m into 5 to the power n, then it is going to be terminating. So why it is going to be terminating is very, very easy. You can see. So let us say you have p by q. Let's now prove it also. So p by q is there and let's say q is equal to 2 to the power n into 5 to the power n. Okay. And so hence what is this number? So this number will become p upon 2 to the power n into 2 to the power n. Simple. So what do we do in this case? Then let us say case 1. If m is greater than n. If m is greater than n. So what do I do? I do this. So I will multiply p 2 to the power m into 2 to the power n into 2 to the power m minus n. This is what I am going to do. Okay. And top maybe you multiply 2 to the power m minus n. This process I can do without any doubt. Any problem because I can multiply and divide by the same number. The fraction doesn't change. But why did I do that? If you see this becomes p into 2 to the power m minus n divided by what will this be? Oh sorry. I have to take 5 not 2 here. My bad. My bad. My bad. Here it is 5. 5. So 5. All are 5. Just change. 5. 5. 5. Okay. Now 2 to the power 5. 2 to the power m. Sorry. And 5 to the power this one adding you will get 5 to the power m also. This one is also 5. This one is 5. Right. Right. Right. Right. Okay. This is this and then what will this be? So this is p into 5 to the power m minus n. And let's say this will be 10 to the power m. Now anything divided by 10 power of 10 will always lead to our terminal decimal number. Is it? This will lead to terminal. Terminal decimal representation without doubt. Why? Because you are dividing by multiple of 10. Right. This is case one. What is case 2? Case 2 is m is equal to n. So if m is equal to n, the number is p upon 2 to the power m into 5 to the power m only. Okay. So clearly here there is no big deal 10 to the power m. Which is again a terminal decimal, terminating, terminating, terminating decimal representation. Third case could be m is less than m. Then what will happen? Then you will write p upon 2 to the power m into 5 to the power n into. Now you will multiply by 2 to the power n minus m. Okay. Multiply divided by this. So what will happen again? You will see this is reduced to p into 2 to the power n minus m divided by again you will see 2 to the power n into 5 to the power n. Right. So hence you will get p into 2 to the power n minus m by 10 to the power n. Isn't it? So this is how you will get all the three forms, all the three cases you will get. Did you all understand guys? Hello. Yes or no? Is that fine? Is that fine with all? Okay. So this is the basic proof. So you can also predict how many decimal points will be there after the point here in case of let's say this first case. After how many digits will it terminate? Can you predict after how many digits will it terminate? m, the larger of the two. The larger of the two. Is it it? So if n is more than m, if m is more than m. Right. So hence let's say if you have 3769 divided by 2 to the power 21 into 5 to the power 25. How many digits after decimal will be there in that expression? Yes. So there will be 25, 25 digits after which it will terminate. Correct. Yes or no? All clear. Both of them. And similarly, so you can this, there will be one market questions like this. So they will give you number of decimal digits after the decimal in this kind of an expression. So one, two, three, five, one, two, nine divided by two to the power 20, 21 into two to the power 20, 22. Sorry, five to the power. So how many, how many will be there? So many people will get confused and they will be like, Oh my God, such a big number. And how to find out if you can. But the answer is simply 20, 22. Right. So it's simple. Okay. Good. So leading to this. So another corollary you can say is P by Q be a rational number such that the prime factorization of Q is of the form this, where M and N are non-negative integer that is 0, 1, 2, 3, 4, all positive numbers as well. Then X has a terminating decimal expansion which terminates after K places of decimal where K is the larger of M and N. This is what is written in statement. Okay. Now let X is equal to P by Q be a rational number such that the prime factorization of Q is not of the form, not, nahi hai. Then X has non-terminating repeating decimal expansion. So any other factor, even let's say if three is included or seven is included or 11 is included, anything but two and five is there. That means it is not going to terminate. So one upon three, classic example, 0.3333, isn't it? So it is not going to terminate because there is no two nor five, right? So presence of this, even this is non-terminating. Even this is non-terminating. So even if one of two or five is there, but the other one is not two or five, but some other prime number, then these are all non-terminating. But since they are rational numbers, so they will have repeating. Repeating. Okay. Repeating. Fair enough. Okay. So we'll see some questions. Typically, one markers are asked. Okay. Here is the first question. Here is the first question. If XY is 180, HCF of XY is three, then find the LCF. So these are one marker, right? So all of you should be able to do what is the concept underlying concept is HCM into LCM is equal to A into B, right? So AB or XY is given 180 and HCF is given three, three into LCM. Can you predict two numbers? Such two numbers X and Y? Who's LCM is 60 and anyone? Can you predict any number? Six and nine. But 16 to nine is not 180. The product is 180. Getting my point. Three and 20. How again? Three and 20 cannot be 180. So it has the product has to be 180. No. Is it? 30 and six. Correct. Only 30 and six or is there anything else? 15 and 12. Yeah. Like that. So that would be. Very good. So good. Next. Next question. Decimal representation of this will terminate after how many decimal places do people don't get intimidated by the numerator, which is there one, four, five, eight, seven. So you don't need to actually start. Dividing. So what is the answer decimal representation of this will terminate after how many decimal four places without any doubt. So you have to basically so the question is this that 14587 14587 divided by two to the power of one into five to the power of four. So after how many places it will terminate. So whichever is larger between one and four. So four is the answer. Perfect. One mark done. This is a sample paper question next one this. So the questions like this also three bells ring at an interval of four, seven and 14 minutes. All three bells rang at 6am when the three bells with the ring. What? When what? When the three balls will the ring together next. Oh, there is some. This is a sample paper issue. So this will be bells. And there's some drafting issue when the three bells build that in together next. Yes. 628 am is the correct answer. Yep. So what do you need to do underlying concept is you have to find out the LCM LCM of four, seven and 14 is clearly 28. I hope you all know the methods of you all know how to find out LCM guys for 714. All of you know, right? So there are multiple methods of doing so I go by fundamental theorem of arithmetic. So four is two square. So prime factor is everything. So seven is seven into one. And then 14 can be written as two to the power one into seven to the power one and then LCM is nothing but how many different types of prime factors. Do you see? So one is two and one is seven. Only two factors are there and write the highest factor, highest of all of them. So in this, this is the highest. So two and seven highest power is one. So answer is 28. Okay. So hence. All right. So hence LCM is this. So six am six plus six am plus 28 minutes is nothing but 628 am. So I have 628 am. All of them will ring together. Very good. So you'll get two marks for this. So four marks for this paper done. This is another proof that two minus root three is irrational given that root three is irrational given that. So this is important. They have already given that root three is good. So how to do such questions you now know. So what to do question number four. So here what you will do is given that root three is an irrational number is rational. So you don't need to prove it separately. You're actually square it. So hence. No. So hence in such questions you say contradiction method. So we have to prove to prove to prove. Two minus root three is also is an irrational irrational number. Fair enough. This is we have to prove. So what how do we say we say let us say let us assume let us assume a or X is equal to two minus root three is a rational number. Let us assume X is equal to two minus root three is a rational number. So assumptions were telling me and we contradict it. That means X is equal to two minus root three. Fair enough. That means you can say root three root three is equal to two minus X. Yes or no root three is two minus X. Now this is good enough for proving good enough. So here is proof hence why because LHS is equal to. An irrational number. Right root three is irrational number given. But RHS is equal to two minus X is equal to a rational difference of difference of two rational numbers. Right because two and X both are rational numbers we have assumed X to be rational. Therefore is equal to a rational number. But how can a irrational number be equal to a rational number so an irrational number irrational number. Can't be equal to a rational rational number hence hence what will impact it will impact our assumption hence the assumption. The assumption that X is rational is wrong is contradicted contradicted. Therefore X is equal to two minus root three is an irrational number. Okay. So you have to write all of this to get three months. Okay. So in NCRT like very similar but instead of taking it as X they've said since two let two minus root three be a rational number. And that means two minus root three is equal to P by Q where P and Q are integers. And then they've proved it using the similar method. No problem. So you can see basically you can do that. Just a minute. Yes guys can you hear me? Hello. Okay. So where were we? Yes. Sorry. Are you saying something? Did I complete my discussion there? So you're saying yes. So what are you saying is P by Q. So let this be P by Q is equal to two minus root three. That's what you're saying. Right. So here you can say that P by Q or two minus P by Q is equal to root three. That means two Q minus P by Q is equal to root three. Again, you will say that this is a rational number. Again, you'll have to prove this. You prove that this is a rational in the left hand side irrational in the right hand side. So hence again, same thing. Right. This is what you meant. Right. When you say that. Okay. So yes, you can do that as well, but whichever logic you think. So I think since root three was given directly. So I assume that X equals to two minus root three is a rational number. So X is equal to two minus root three. So root three is equal to two minus six. We are adopting the same concept that. So I am assuming the concept that rational number difference is a rational number here. In this case, you'll say integer by integer is a rational number. So adopt this method and say I do suggest this. Okay. So let me let us solve a similar question. Right. In this method. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. Okay. So the factors of a prime number is. So two. Right. So two factors are there. Total number of factors are number one and so be careful. Easy questions can be tempting. Okay. Next. Now it's a FNLCM of 122115 respectively are. So here's the question where you have to find out. It's a FNLCM both. Right. So 12 can be written as two square into three to the power one. 21 can be written as three to the power one into seven to the power one. And 15 can be written as three to the power one into five to the power one. Is it? So hence it's CF. It's CF. Right. All the factors that you see. So two into three into five into seven, but put the lowest of all power. So lowest of all. So lowest is made. It's my lowest. Two color west is zero. Why? Because here is two to the power zero. Here is two to the power zero. What is lowest of three? One. What is lowest of five? Zero. What is lowest of seven? Zero. Why? Because here it is five to the power zero. Here it is seven to the power zero. So everywhere you fill it. So it will make sense. Right. So hence the, this thing is three. What is else here? Right. All the same factors two into three into five into seven. But now this time put the highest one. So highest of two is two. Highest of three is one. Highest of five is one. Highest of seven is one. Is it? So hence whatever it is. So two, four, three, 12, 67, 420. So the answer is three and 420. That is C. Okay. Perfect. So. Okay. The sum of exponents of prime factors in the prime factorization of 196 is, so this is again based on FDA. So some of exponents of prime factors in the prime factorization of 196. So 196 is two times. So you have to do like this. This is your typical method. Two into 98. Two into 49. And this is seven into seven. So hence 196. Is equal to two square into seven square. So what is the answer? Two plus two four. Right? So exponents you have to add. So answer is four. Okay. Next. Prove that root five is an irrational number. Every alternate year this question comes. Either it will be root five, root seven, root two, root three, something like that. Prove that root five is an irrational number. So what do you do? So again, here is what we had discussed and we are going to use that part. What will you say? So we'll see a question. So prove that root five is an irrational number. Irrational number. That means it's decimal representation will be non-terminating, non-repeating, but how do we prove that? So we do again by contradiction method. Okay. Let us say, let us assume root five to be a rational number. We can do that. We can start with this assumption. So where P by Q is equal to root five, you say, where P and Q are co-primes. There is no common factor between P and Q, P and co-prime integers. And you say co-prime series, but co-primes integers where Q is not equal to zero. So all the definition of rational number you have to write over there. Yeah. Just a minute. Someone is constantly dropping off then joining Debashish. Are you there? Can you hear me? Debashish, what's happening with you? So there are, there is only one Debashish or there are multiple Debashish because I'm seeing someone is joining, dropping, joining, dropping, joining. What is happening? Is the internet connection stable? There is some issue. Are you the same Debashish or some other Debashish is there? Just, so because I can see constant, you know, someone is attempting, but you're not able to join properly, I believe. Anyway, so now P by Q is root five where P and Q are. So what do we do now? So we say that P is equal to root five Q. Okay, so P square will be equal to five Q square. That means you will say, what will you say? Five divides P square. Correct. Now in five divides P square, then five divides P. Remember, we prove this that if a prime number divides A square, if a prime number divides A square, it divides A as well. We prove above. Isn't it? So five divides P. That means I can write P is equal to five times M for some, for some positive integer, positive integer M, right? Because five divides P. So P is five times some number M. Now, so now come again. So P by Q was equal to root five. Okay. And or you can start from here only so you can just mention this number. So let us say one. And let us say two like that. And then you say from one and two. From one and two. What can I say? I can say five M whole square is equal to five Q square. This means 25 M square is going to be equal to five Q square. That means Q square is equal to five M square. This means five divides Q square and five divides Q. Correct. So four P and Q both have a common factor. Common factor. What common factor is this? Five. But boss, you said that they are co-primes. You started with that. They are co-primes. How can co-primes? I hope you understand what co-op. What a co-prime number. What a pair of co-prime number is. Co-prime numbers are those numbers, which has only one common factor. That is one. Okay. There is no other common factor apart from one. So hence they are, you know, so they cannot have five as a common factor. So then, you know, it contradicts our statement or it contradicts. It contradicts the assumption. What was the assumption, guys? The assumption was it contradicts the assumption that root five. Sorry. Assumption that P and Q are co-primes. Hence root five is not a rational number. So this is how you have to do it. Sir, is there a pair of co-primes, which are independently composite? Yes. 9 and 10. Every adjacent to adjacent numbers are always two consecutive numbers are always co-primes. See, 9 and 10, both are composite, but they are co-primes. Similarly, 14 and 15, co-prime. Similarly, 15 and 16, co-prime. Similarly, 20 and 21, co-prime. So all are. Okay. So two consecutive numbers are always co-prime. Now, this is done. Very good. You tell me. HCF of 135 and 225. Chinnu unnu question. Chinnu unnu. Answer is one marker in 1920 board. So the board exams will be so predictable. You will make mistakes because you are bored. That's the only. Out of boredom, you will make mistakes. But what is the answer, though? So 135. Either you can prefer long division methods. So 135 is a little low. Is a little low. 225 is a little low. So it is one, 135. And either Kavathika 90. Right. Either for a second low. 135 because this is the Euclid's division algorithm. 91, 90, 45 and 90. So the last divisor is the HCF. Okay. This is by Euclid's division algorithm. So I have not written in the typical form. The long division method is nothing but division algorithm. Otherwise, you could have done this. 135 is equal to 15 into 9. So 27 and 5. So 3 to the power 3 into 5 to the power 1. 225 is 5 square into 3 square. Isn't it? So now HCF. How to write HCF guys? HCF KLA. How many factors you see? All the factors. Write them down. So 3 into 5. Like that. And the lowest power is amongst all. So out of 3 power 3 into the lowest is 2. Out of 1 into the lowest is 1 to write 1. So 45. This is another method. You know, FTA route. This is called FTA route. Okay. So very good. Next. The exponent of 2 in the prime factorization of 144. Again, a chiller question. Very, very sad. No, exponent of 2. 2 to the power 4. Yes. 16 into 9 is 144. So, so these questions you should not make mistakes at all. So 144 is 2 to the power 4. 16 into 3 square. 16 9s are 144. Understood. So how much is it? Answer is exponent of 2 in the prime factorization of this. So 4 is the answer. B. Next. This kind of a question. Prove that root 2 plus root 5 is irrational. But it's not given. It's not said that, you know, root 2 and root 5 are rational. Irrational. What will you do? Will you go and prove root 2 and root 5 independently? Should we write the only option? Now this year we will not get any MCQ, I believe. But it's always better to write the, you know, just in case you are making a mistake. So maybe you can get some marks. So write the option and the answer. How much time does it take? I will tell you what has been instructed from your school. So have they told you to prove them independently and then go for the sum? Or you assume already that root 2 and root 5 are irrational. Assume. Okay. What has been told in NPS guys? Prove them independently. What so? So for us, they've told if it's less, if it's one or two marks, then we can assume if it's three or four marks, we have to prove them. It will be real long then. So, you know how to prove root 2 and root 5 are irrational. But we just prove that. So I will do the last part of it. So I will suggest. Yes. This year, so there is no four marks. But if it is really a five marker question, then obviously you can go for it. Proving them independently because I would say even in three marks, you can just bitch it. So you can't, you don't need to, in my opinion. If it is five marker, then prove root 2 separately, root 5 separately and then understood. So this is this year. So there is no four marker. No, so that way is your safe. And anyway, there are three, five markers. So you will be having time to do that. But anyway, so let's say we have to prove assuming that root 2 and root 5 are irrational. So you can start with assumption that let root 2 plus root 5 be a rational number, be a rational number. So you can say that then that means it can be expressed as P by Q is equal to root 2 plus root 5. You can do that. Is it P by Q is equal to root 2 by root 5 where you have to write where P and Q are integers. P and Q are co-primes. Do not forget to write this statement. This is the soul of the entire proof. And this also Q not equal to 0 in a rush in the rush in the hurry of completing. So you might miss the three points integers, co-primes not equal to 0 all have to be written must. Okay, now squaring both sides. You have to square in this case, because any which any ways you love to square, whether you transpose one of them into this side and then do the squaring whichever you have to square squaring both sides. In that case, what I'll suggest is do not just blindly go for proving root 2 and root 5. So that was you please first make sure what you need squaring both sides. You will get P square by Q square is equal to 2 plus 5 plus 2 root 10. So in this case, you don't need to prove root 2 and root 5 separately. Just by proving that root 10 is a prime is an irrational number you are done how so you can say P square by Q square minus 7 is equal to 2 root 10. So that means you can say P square minus 7 Q square by 2 Q square is equal to root 10. What is this? We need to square root 2 is not possible. I don't think we need to square root 2 is equal to a minus root 2 by B a minus root 2 by B. Okay, which is not possible. What are you saying? Sir, I meant to say that it's given that root 2 plus root 5 will be P by Q. So if we take root 5 to the other side, we get P minus Q root 5 by Q is equal to root 2. So root 2 is being expressed as P by Q which is not possible. No, there the criteria is that P and Q both have to be integers. But the moment you are writing Q root 2 integer identity is gone. So it cannot be just any P and Q. It has to be integer Q has to be an integer. These are integers. So when you understood, so that root will not help. So what you're trying to say is this I believe what I understood is this. You are saying sir write P by Q minus root 2 is equal to root 5. Correct? Yes, sir. And then you write B minus root 2 Q by Q is equal to root 5. Yes, sir. And then you are saying now again it came into P by Q forms and how can it be no. Why? Because P minus root 2 Q is not an integer. No. So but he doesn't say that root 2 is irrational. We don't know that. So as far as we are concerned P minus root 2 is also an integer. So how do you know? So hence anyways, you are assuming that root 2 is also a rational number then. So it doesn't say it's irrational. So that doesn't prove to what we need to assume that it's irrational. No, no. I'm sorry. I didn't get the logic. I'm saying it doesn't say that is not see it is talking about root 2 and plus root 5. It is not independently talking about root 2. Is it? So how do you know that this part is an integer? So just because you don't know what you're multiplying with. So let's say if you have a X minus 40, is it an integer? If I generally give you X minus 40, you can't say because you don't know what X is. For X minus 40 to be integer X has to be an integer. 100%. Right? X could be a complex number here. So there is no point of assuming that X minus 40 just because X is not given. Yes, if I say that X is an integer, then you can say X minus 40 is also an integer. But if I don't say X is an integer, can you assume that since it has not been given, what is X? So X minus 40 will be an integer. Yes. Get the logic. Yep. So don't assume without the, you know, so there's a term in, you can't assume by, you know, losing the generality part of it. So you can't just say like that, right? Okay. So there has to be a logic behind it. So I think all of you are clear now. So here is, this makes sense. This makes sense. Why? Because LHS, what is LHS numerator? Numerator is an integer. Is it how and why? Because it is nothing but p square, which is integer square minus seven times another integer square. So integer minus integer, integer. No problem. And denominator q square, which is integer into integer square is equal to integer. No problem. So hence we see that the LHS is integer by integer form, integer by integer form, form, and clearly denominator is not equal to zero. Denominator is not equal to zero. That means root 10 can be expressed as a rational number. Correct. But you can disprove it by doing what? Why can't they be real? What is that? Why do we assume p and q to be integers? Why can't they be real? Oh, my God. So this is the basic understanding. So that is the definition of rational number. How do we define rational number? What is the definition of a rational number? Why did we require rational number for that matter? We were not sure how to divide nine breads in 10 people or 10 breads in nine people. So that was a genesis. So we were not very clear if you have to, let's say there are 10 people or nine people claiming 10 breads. So how do we... Yeah. So hence that is the first question. So understood. So hence rational number is defined as integer by integer. First of all, where the denominator cannot be zero because that expression is not defined and it has to be reduced form in the reduced form. That means there must be co-prime. So p and qb by q where p and q are integers. This is what is the definition of rational numbers, right? So p and q have to be integers. q cannot be zero. And p and q, gcd, that is greatest common divisor or hcf of p and q is one. Or they must be co-primes. That is the definition of rational number. So hence we have to... If you're saying something is rational number, you can define like that. And that's what I started with. So I assume that it's a rational number. So by definition, it should be like this and built by proof. So actually I got what? That p square minus 7 square by 2 square is root 10. Now someone is saying something. Arjun, yes Arjun, unmute and say, sir just get root 5 to the other side and then square. Didn't understand. Once again. What is that? So I can say root 2 plus root 3 equals to a by b. Like co-p by q. Root 2 plus root 5. Yeah, root 2 plus root 5 is equal to a by b. That's what I did, p by q. And then you get root 2 to the other side. 10 square. That's fine. You'll get the same thing. So anyways you'll have to prove either of them. Either root 2 is an irrational number or root 10 is an irrational number. If what you say? Understood. So what Arjun is saying is instead of doing all this, you do this p by q minus root 2 is equal to root 5 and then square. Isn't it? That's what you're saying? Arjun? Am I right? Now you square it. Square it. No problem. p square by q square plus 2 minus 4. Sorry, 2 root 2 p by q. Correct? Is equal to 15. Sir, I guess root 2 is irrational but that contradicts the fact. That's what I'm saying. So anyways, that means you are assuming that root 2 is irrational, isn't it? Yes, sir. So the moot point was if it is a 5 marker question, then you have to anyways prove one of them as irrational, isn't it? That's what I was saying. So what will you write here? You'll write p square by q square minus 3 is equal to 2 root 2 p by q. So you take everything on the other side. You'll get q by p by 2. Okay, into p square by q square minus 3 is equal to root 2. You have to segregate these parts. So clearly this is integer operation. So this is our rational number. All our integers, q, p, all our integers. So LHS is a, clearly LHS is an integer. But RHS is I mean a irrational number. Sorry, LHS is a rational number. Clear? It's a p by q form. But this one you have to anyways prove. If it is given, then no problem. If root, if it was given that root 2, that's what I'm saying. Since root 2 is not given now from here, the next step of the proof is prove that this is irrational. Root 2 is irrational. So you'll stop here, then you'll go where? Let's say root 2 is equal to p by q. Correct? So this means that root 2 is rational. So if root 2 is rational, so this entire thing can be written as let's say capital p by capital q for whatever p and q are. And then again, build your proof to prove that root 2. It contradicts that root 2. Yes, that's what I'm saying. If many schools have given different instructions that if it is a five marker, then after this, don't stop in RSS or whatever you are mentioning. They will stop here assuming that root 2 is irrational. But if it is a five marker, that's what I asked you in the first place. What has been instructed to you? Because you have to follow the whatever has been. So hence, in my opinion, you should stop here, leave some space, and go back and solve other questions if you're running out of time. Then let's say if two, three minutes are left or four, five minutes are left and you're revising, come back in that free space, complete the proof and, you know, place it. Did you understand? So you can adopt these, but then you have to be very cautious that you're doing and you have to mark somewhere that I have to come back to this question. Okay, fair enough. So this is how you should be doing it. This one. Oh, cool. Two plus root five by three is, what is the answer? Is, is, what is the answer? Rational. Two plus root five by three is rational number. Two plus root five by three is an irrational number. What if I write a real number? Will it be wrong? Two plus root five by three, real number or not? Real is more correct. Every non-complex number will anyways be a real number. But here the answer expected from you is irrational number. So you have to write irrational. Okay, so you have to answer in either rational or irrational term. So clearly it is an irrational number. Why? Because some of a rational and an irrational number is irrational. Okay. Always. So let's go to the next one. This we did already. So in one set, they are asking you to find out the HCA in the other one. They're asking you to find out the LCM. Find the LCM. Same year board paper, two marks. So underlying concept LCM-HCF is A into B. Oh, my, this thing is running out of power. The answer is answer is people are why are people dropping off and joining in so many times? Okay. So LCM of any two number is given by A into B by HCF. So be careful while you are writing the calculations. So this is 135 into 225 divided by 45. Isn't it? So this is the thing. Now this is five times. Correct? Yeah. So simply it is 567. Okay. Easy one. Two marks. How many decimal places will that decimal representation of the rational number this terminate? The maximum of the two. Okay. Next. If the LCM of two number is one of your similar questions, can you see that? Typically similar questions. If one of the numbers is 265 the other. Yes, very easy. Again in this question also. So 182 and this 13 is the HCF divided by 26 will be the answer. So twice 91. So 91 is the answer. 91 is the answer. Very good. C. Now this is this has been already given five plus two root seven is an irrational number where root seven is given to me. So you don't need to prove that root seven. So hence you will say what? P by Q is equal to five plus two root seven. This is what you are going to do. Isn't it? So you will say P by Q minus five is equal to two root seven. That means you will say P minus five Q by two Q is equal to root seven. And you reach the same level. Reach the same level. Isn't it? So this one is rational number and this one is irrational number. Done. Next. Find a rational number between root two and root three. Find a rational number between root two and root three. This is 1819 the year before. They are asking you to find a rational number between root two and root three. What was the value? Yes. Friends. You don't need to write such a big number. Simple one. Is one the right answer? One. Is one between root two and root three? Root two point two five. Just write one point five. Rational they are asking. Rational number. One point five. Four. Because root two is one point four one. Root three is one point seven. You don't need to write such long numbers. Yes. Why do you need to write such long numbers? Why are you wasting time? Find a rational number between root two and root three. Find an irrational number between root two and root three. If that the question is find an irrational find an irrational number between root two and root three. Can anyone tell me? Irrational number between root two and root three. How will you calculate that? Root two plus root three by two. Perfect. Very good. Or fourth root of six. How? Any number. If there are two numbers A and B then A plus B by two will be between A and B. Right? Is it it? Hence root two less than root two plus root three by two less than root three. So clearly this is an irrational number only. So that's correct. Or if you can just generate one. So you know 1.5 is safely between 1.4 and 1.4 1.41 and 1.73. So you can write 1.50 1 0 0 1 0 0 0 1 0 0 0 0 1 like that. You can create any number. Or another way of this thing is square root of root of two, root of three. That means six power one by four. This is also an irrational number between root two and root three. This is called geometric mean. So people whom which we discussed MGM. So this is geometric mean. So geometric mean is also between the two numbers. So this is also six power one by four. Will be between root two and root three. Irrational number. Anyways. Next. Similar. The most basic one proves that root two is an irrational number. We are not going to do this. Next. This one same thing again. One year before. Same question. Only numbers have changed. So you now know. You can predict what all questions are coming. So if HCM is five LCM of the same numbers. Just be careful if the numbers mentioned are same. Many a times they will you know. So you will you know you are so overconfident that you know this is the type of question one. So there they can they can give you three thirty six fifty four and here three three thirty six sixty four and then do boards are not notorious for this but then you never know what is the answer. So hence the answer is three thirty six into fifty four right divided by six so six into nine right. So nine three zero two four is the answer. Okay. Good. Next. Same. Same thing. Another previous year. Same question. Same as same pattern. Two plus five root three is an irrational number. What will you do? So many times you would have done this by now. So P by Q is equal to two plus five root three. So there it was five plus two root three. Now they are saying two plus five root three. Amazing. So P by Q minus two is equal to five root three. So P minus two Q by five Q is equal to root three. So in three step you again land up in the same this thing. So this is integer by integer form. This is irrational number. So this is Rn. This is In. So not possible. So this is the basic of this one. Let's go to the next one. Hmm. Find after how many places of decimal the decimal form of the number will terminate. What is the answer? Everyone please please please please Aditi says four. No. Who says four? Four. What is the answer? No. How do you say four? How are you saying four and all? How? How? This thing is divided by nine. No. So it has to be only two and five. There is no scope of three. Only two and five should be there. Understood. So it is a non-terminating. Oh, that is correct. Sorry. Sorry. Sorry. Yes. Yes. My bad. Yeah. My totally bad. I did not see 27. Fantastic. Good. Good catch. Yes. Yes. 27 is yes. But let's say it was one upon. I was thinking of one upon. Yes. Very good. Yes. Four. Correct. Correct. Correct. Good. Good. Good level of concentration. Nice. Closing. Close. You will be rewarded for this. Good. So express 429 as a product of its prime factors. Is it a prime number? So this is one marker again. Is it a prime number? 429. Clearly no. It gets divided by 3. So what are the factors? 429. 429 is 3 times for sure. So 3 and what is this? 3 into 1, 4 3. So 1, 4, 3 into 3 and 1, 4, 3 itself is going by 11. This is 3 into 11 into 13. 3 into 11 into 13. Perfect. Cool. So next one. This is a word problem. You can expect these kind of questions also. On a morning walk three persons step out together and their steps measure 30, 36, 40 centimeter respectively. What is the minimum distance each should walk so that each can cover the same distance in complete steps? So what is the minimum distance each should walk so that each can cover the same distance in complete steps? So basically they are asking you to find out what LCM of these three numbers so that they can complete that distance in integral steps. So LCM of 30 36 and 40 in centimeter. So 30 is equal to 3 2 into 3 into 5 36 is equal to 2 square into 3 square and for my sake I will also write 5 to the power 0. 40 is 2 3 into 3 to the power 0 into 5 to the power of 1. So prime factorization. So hence LCM is write all so 2 into 3 into 5 and the highest power so highest power is 3 for 2 3 highest power is 2 5 highest power is 1 ok those. So 40 into 9 360 centimeter. So after 360 centimeter they would have so but you have to find out minimum distance each should walk. So 360 each should walk so be careful if they are asking minimum number of steps each should take to how many number of steps first person will take 12 steps the second person will take 10 steps and the third will take 14 so 9 steps so in 12 steps 10 steps and 9 steps together meet so just be careful if they are asking about that. Next similar another type of word problem find the largest numbers which on dividing 1251 9377 15628 leaves remainder 1 2 and 3 respectively I don't think they will ask you because it involves again that lemma but so be prepared so how do you solve find the largest number which on dividing this and this leaves remainder 1 2 and 3. So basically in such questions so the largest number is X let's say so 1251 is equal to X into let's say Q1 plus 1 so 1 is the remainder and the second is 9377 and hence I said this is lemma question so X is the same then next quotient is 2 and the remainder is 2 and 156 3 sorry 6 28 is equal to X into another quotient plus 3 right so hence what will you get 1250 is equal to X into Q1 then 9375 is equal to X into Q2 and 15625 is equal to X into Q3 so basically they are asking you to find out the highest X which is nothing but HCF of all 3 HCF of these 3 numbers 1250, 9375 and 15625 okay so 1250 appears to be easily 5 cube into 5 to the power 4 so it is 125 into 10 so 625 into 2 so that is 5 to the power 4 into this is 1250 what about 9375 so you can do the now for 9375 again it definitely goes by 25 so let us write like this 25 into 25 3 is 75 and 18 so 187 so 7 is 87 right so wait a minute 187 so 187 means 12125 so into 5 so this is again it will go by 25 again so 25 into 25 into this is 1 and 12515 so this is the thing okay those so clearly you can see some trend coming in so larger numbers be very very careful so 625 definitely is going to be divided by 6 15625 again so 15625 it will clearly go by 25 so 25 6 times so I will write it here 6 times 150 so 62 means 2 and 125 is also 625 into 625 this number is fair enough so you got enough of further now so this is 5 to the power 4 into 2 this one is 5 to the power how much 2 2 and 1 5 into 3 and this one 4 and 2 6 correct so the hf is 5 to the power 4 625 so this is the answer okay those so this is what 625 I hope everyone is getting the same answer did you get the same answer guys 625 yeah 625 yep below correct you can check also now below any doubt anything any problem those two hello e5 to the power 5 is 625 into 5 correct problem yeah next question this is another one marker this is 2 years prior to 1718 for 2018 board what is the hf of smallest prime number and the smallest composite number answer is only one answer yes 2 right 2 is the answer what is the smallest prime number 2 what is the smallest composite number 4 so smallest prime smallest composite don't say one is composite number many people make this mistake smallest composite so hf of 2 and 4 this is again same thing just one year before and these are all actual board paper given that root 2 is a rational number they have only changed 3 code 5 plus we had seen what let me show you this 2 plus 5 root 3 here and here 5 plus 3 root 2 numbers change but root 2 we have done or pallet or pallet 2 plus root 3 this is root 7 so they have every year this kind of a question is coming somewhere 2 minus root 3 so every year this is typical of there will be one question for sure ok now this find hcf and lcm off and verify so you have to verify also find hcf and lcm off 404 and 96 and verify so lcm of 404 so thankfully this is a good number 2 square into 101 and 96 is 2 square into 49 that is 7 square 101 is a prime number so what is the hcf 4 hcf is clearly 4 what is lcm very big number lcm will be 4 2 1 0 1 yes or no 404 into 96 by 4 so 101 into 96 sorry what happened 404 is right what is the problem 96 it is 2 to the power 5 into 3 I thought 98 2 to the power 5 into not this correct good good good thanks so so now what is hcf hcf is 2 to the power 4 hcf is still 2 to the power 4 lcm is sorry 2 to the power 2 lcm will be lcm will be 2 to the power 5 into 101 into 3 do not overwrite here cut it and then together is this the lcm correct so multiply this so 303 and what is this 32 into 396 into 101 so 96 96 correct 96 96 00 96 correct clear so this is so what type we saw so if you summarize you will be getting questions on terminal decimal terminal terminating terminating terminating decimal take one marker two marker maximum one or two markers so you know maximum of the two so they will ask you question in terms of which are having terminating decimal or after how many digits they will terminate questions like that this is on fundamental theorem of arithmetic and then we will questions on lcm hcf lcm into hcf is equal to a into b questions are on these third is proving that prove that root p and combination whatever combinations they are giving combination are irrational these are the three types of questions you will get this will be a little higher mark where p is a prime so hence questions like root 2 plus 5 root root 2 plus 5 or root 2 minus root 7 all that you know combination of that so this will attract say let's say 3 to 5 marks there is no four markers so 3 to 5 markers this will be around root 2 marks and this also will be around 1 to 2 marks so you can expect this in the world okay now this is how see smallest prime smallest composite this is how they have done so hcf of 2 4 is 2 and they have written also hcf of the smallest prime smallest composite is 2 you don't need to write in one marker you can just now given root 2 is irrational prove that 5 to prove 5 plus root 2 is irrational so let me just yeah this is how they have done it so let us assume 5 plus root 5 root 3 is rational so it is in the form a by b or b by q where the condition has to be mentioned a by b belongs to z either you can write like this b is not equal to 0 and hcf a and b is equal to 1 so you can write in words also that a and b are integers or p and q are integers q not equal to 0 and p and q are co-prime you have to mention that if you don't mention this statement then you will lose marks now 5 plus root same thing that we did transposing and this shows that root 2 is rational a minus 5 b and 3 b are integers but we know that root 2 is irrational this contradicts our assumption that 5 plus 3 root 3 is rational so hence 5 plus 3 root 2 is irrational hence prove this is what 2 marks all this effort for 2 marks okay this was 3 marker 404 and 96 question to find hcf and lcm so you please do whichever way you are most comfortable so let's say this is branch tree structure you want to do please do that I should have also done this because this will eliminate errors so don't try to do it in mind like what why was trying to do right isn't integer represented with i no integers are integers are represented by the set z z z is something like that you will see that it is German this thing for you know integers so hence z comes from the German word so rational a complex numbers are given by capital C the set the set of so natural numbers is n whole numbers we generally do not say we say non negative integers then integers are z then rational numbers for q irrational number for i and real numbers are and complex numbers see these are the usual notations for different sets of numbers yes z is taken as complex number inside like let z be a complex number so for that matter we take even q as an integer isn't it so that these are normal conventions okay so 404 2 square into 7 into 30 96 is 2 to power 5 into 3 hcf is greater scum factor 2 square is 4 lcm all factors so least power so like that 9696 product of 2 numbers this verification part you are going to do so product of 2 numbers 96 into 404 product of hcf into lcm is same amount so hence hcf lcm is product of 2 numbers so in verification don't try to copy the same number we will verify because it will act as a check also for your calculations so that's how you have to do perfect so we now come to the closure of the revision program for mathematics so we have touched upon all the topics i think if anything is have we not touched anything so with that we have some time we can do that else i think most of i think every topic we have touched upon all the ppt's and the videos will be available as i told you more than 50% of them are already available the rest of them we will add it and put it on the web so you can know your strategy ideally should be to revise in a systematic manner from here on till you write your boards so that will give you a good that will keep you in touch with the topics parallely do not do only this till boards because that will be a lot of wastage of time so hence in my opinion you should all yesterday was kbbi exam i don't know if any of you anyone of you could you know get to see the type of questions and all so it will be a good target next target should be that whether you want to get into isc or not preparing for kbbi gives you a good good boost or you know let's say acceleration towards the next two years so when you have an objective in mind it becomes you know much helpful so hence now you are kind of done anyways in the month of february you will be writing another set of pre-boards so that will give you a good you know what you say confidence and you eventually become bored so hence instead of that happening you start with next goal start working on it anyone requires any assistance apart from this also please touch base with me individually i will help you and you guys can start on that journey so with those words i think i can leave you early today if anyone has any doubt or any question regarding boards or regarding preparation for anything you can ask now so you can discuss those next steps anything yes anyone anyone wants to discuss anything i hope these are how are these classes guys so how did you find them so all the revision classes which we did because we are anyways planning to repeat it for multiple number of matches again so not maybe not if not this year next year so what's your take so did did these classes help you was it good or what was your this thing so i would like to hear from you you can just drop in a text individually to me as well or if you want to share in the comment chat box so i i get to know how was how was your this thing so yes anyone wants to share his or her now you can write to me so do let me know so how did you feel about these classes and what else can be done so that your performance in the board exams can be improved so do let me know okay guys with those words i think any if you anyone of you has any other doubt you can just ask me refreshed right okay great thanks thanks are in for those words and i would request all of you to go go through these once again so you can just revise the concepts now so fair enough thanks for all your time and my best wishes to all of you and if you are you know will try to again arrange anyways you know that we are back with our next goal so hence that will be there and any other assistance you want me to assist let me know okay thanks thanks for those words thank you bye bye guys take care and all the best from my side bye bye