 Hello friends, in the sessions we have done so far, we have discussed Euclid's division lemma. We saw the proof of division lemma. We also saw some practical examples of division lemma. So today it will be an extension to that session and we'll understand the practical aspects of division lemma and how in mathematics we are going to use it, especially in number theory. So just to, you know, revise a division lemma, we know that if a and b are two integers and I know on a and b are positive integers and we can express a as bq plus r where we know that q and r are also positive integers and the condition is 0 is less than equal to r less than b. So this is what division lemma says. Now this is a very good tool to actually express different types of integers, especially the natural numbers. So if you see first categorization is odd and even. So, you know, we'll use our, you know, division lemma to categorize natural numbers into odd and even numbers. Now if you see any number if you can, if you take and you divide by two, that means my divisor here is b which is two. So if b is two, so any number a, any number a can be expressed as b that is two. So two into k plus minus plus one or two into k plus zero. What does it mean? So if you see any number you take, let's say five. So five can be expressed as two into two plus one and let's say six. Six is two into three plus zero and then seven. Seven will be expressed as two into three plus one. Eight will be expressed as two into four plus zero. So hence any number, any natural number you take the choices of remainder are only two. Why? Because we know that zero is less than equal to r less than equal to b. Now in this case b is, b is two. So hence we will have what? We will have zero is less than equal to r less than, less than two. Now r, which is an integer and it is less than equal to zero and less than, sorry, greater than equal to zero and less than two. Only two possibilities of r are there. One is zero, another is one. Okay. Now here we are going to see how we can express any given integer in form of multiples of three and something added to multiples of three. So let's say if I have seven, I can express this as three into two, three into two plus one, isn't it? If I have eight, eight can be expressed as three into two plus two and nine can be expressed as three into three plus zero. And similarly ten can be expressed as three into three plus one, eleven can be expressed as three into three plus two and twelve likewise three into four plus zero. So if you see the remainders are repeating themselves. So after one, then two, then zero, then one, then two, then zero. Right? So if in language of Euclid's division lemma if you have to write, this is what? This is A. This is my B. This is Q and this is R. So clearly here Q and R integers, Q and R are integers. I can write it like that. And R is greater than equal to zero. So zero is also a possible value of R, but it is never going to three. It is, it is less than three. So the maximum value of R is two. So that's how. So if I have to generalize any integer A can be expressed as three K or three Q, whichever way you want to write it. A is equal to three K plus zero or A will be of the form of three K plus one or A will be of the form of three K plus two. You can see A here. A here is, if you see, this is the case where this is seven. So if you see, now, sorry, this one is six. So this is nothing but three into two plus zero. And this is seven. This is three into two plus one. And this is eight. And this is three into two plus two. So any integer now, whether it is six, seven, eight, nine, 10, 11, 12, 13, 14, anything, you can express this as either of the three cases. K could be anything. K could be any other integer, but then all of them in, for example, let's say 25. So 25 can be expressed as three into eight plus one, where eight is K here, let's say 31. 31 is three into five, sorry, three into 10 plus one. Yeah, let's say 73. This is three into three into 24, 24, that is 72 plus one. Right. Let's say 89. 89 is three into 29 plus two. Yeah. This one also be expressed as and also be expressed as a three K plus two can also be expressed as three K minus one, where K is, don't get confused that these two K's are same. This is just a variable for any particular integer. So hence, so hence, if you see 29 can be expressed as three into nine plus two, as well as it can be expressed as three into 10 minus one. Right. So nine is K here. In this case, 10 is the value of K. Correct. This is for three. Now, what about four? Let's say if you have B is equal to four, then what happens? Now, any number, let's take another example. So let's say four, five, six, and seven, these are different numbers we are going to try. So four can be expressed as four K, that is four into one plus zero, five can be expressed as four into one plus one, six can be expressed as four into one plus two, and seven can be expressed as four into one plus three. And the moment you go to eight, what happens four into two plus zero. So hence, repetition starts. So zero, one, two, three, these are the possible values of remainders when an integer is divided by four. So hence, any number can be expressed as a is equal to what four K plus zero. It can be also expressed as four K plus one, or it will be either of the form four K plus two, or it can be of the form four K plus three, which can also be written as, if you see, this can be also written as four K minus one. So don't get confused again. This K and this K are not same. This is just a variable to express, right? So any number, any number, any integer for that matter can be expressed as either four K plus zero or one or two or three. There are four possibilities. So this is possibility number one, possibility, possibility number one, possibility number one, what all numbers fall in this category. So all multiples of four are falling in this category. So this is possibility number one, possibility number two, number two will be what? Five, nine, thirteen, seventeen, likewise. And number three, when the remainder is two, you'll get six, ten, fourteen and eighteen like that. If you divide six, ten, fourteen, eighteen, you'll get remainder as two. And other possibility is number, you know, when the fourth possibility is number seven, eleven, fifteen and nineteen. So if you divide these numbers, you will get remainder as three, right? So four possibilities. If you go ahead, if you see eight, now eight comes here. Eleven plus one twelve, twelve comes here. So it will be of the possibility one. So these are the four possibilities. Now you can generalize it. It doesn't matter whether you take B as three or four or two or whatever. If you have any B, so you now know any value of B, how can we express? For example, let's say B is 99 or B is, yeah, B is 99. Let's say B is equal to 99. So any number, any number, any integer, again, whenever I say number, it means any integer can be expressed as what? So any A can be expressed as 99 as 99 into sum integer k plus zero. Then 99 into k plus sum into plus one. Then 99 into k plus two. So different, different numbers will be of these, these possibilities. And finally, how many are such possibilities? So finally, it will be 99 into k plus 97. And 99 into k plus 98. I cannot have 99 as a remainder. If that happens, then again, it goes back to this. If it is 99, then you know, I can take 99 common and it becomes 99 times k plus one. So it is as good as this. So remainder will be zero again. So how many possible values of remainders? I have 99 possible values of remainders, 99 possible, possible values, values of remainder, remainder. If any, any random integer is divided by 99. So you will never get a remainder of one or two, you'll always get a remainder of either zero, one, two, three, or at max 98. Right? So this is what, you know, you'll be encountering a lot of problems where expressing one number in terms of a divisor and its remainders will be necessary. We'll see that in the problem solving videos, which are going to follow this video. Thanks a lot.