 Hello friends, so in the last session we discussed about the meaning of Euclid's division Lemma. We tried to understand the meaning of each of the words in this phrase that is what is the relevance of Euclid, what is lemma and division we anyways know. So we understood that lemma is a supporting proposition which helps in proving something. Then Euclid was the Greek scholar who actually gave this lemma somewhere in 3rd century BC and he has mentioned this lemma in his book called The Elements and it's actually in the volume 7 in which it appears. Today we let us go and understand what exactly this lemma mean. So we will start with our understanding of division. So if you remember when we are playing with integers let's say and guys please remember that we are only talking about integers we are not getting into the realm of ratios fractions and all that. So we'll always talk about integers. So whenever we have two integers and let's say when we try and divide one integer with the other we know that in the division process we will have one quotient and one remainder. Isn't it? Let's say I have 25 and I want to divide 25 by 7 so you know the division long division method. So if you see 7 3 times gives you 21 and hence 4 is remainder right and then the moment we get a number which is less than 7 we stop there. So you know that this part or this number is called the quotient and this number is called the divisor which is dividing so hence divisor. This part was called as dividend and you know this part is called remainder. So these are the four terms usually which we have encountered while doing long division method. Now what is lemma in this case? What is meant by or what did Euclid propose in this lemma? So Euclid said now whenever you take any two integers let's take an example and then understand it properly and then we'll go to the definition part. So he said you take any two integers let me take now in this case 15 and other integer let me take as 55. Okay so I'm taking these two integers so let's say I call it a and b so let's say this is a this one is a let's say and this is let's say b okay so these are two integers where b is 15 a is 55. Now when I try to divide 55 by 15 do you think I'll be getting a remainder as 0? So clearly our understanding of division says that we will not get 0 as the remainder in this case why because 55 doesn't lie or doesn't find the position in the tables of 15 right so anyways if we go by the standard long division method and try to divide 55 by 15 we'll get 15 times 3 is 45 and we get remainder as 10. Now can we go further and divide it in if we are restricting ourselves to the realm of integers then we cannot because then if you see the moment I get a number which is less than 15 I stopped this is what we have learned so far so the same division method you could express in a different way and if you see we can write the same thing as 55 can be written as now so dividend so as 55 is a dividend 55 can be written as 15 times 3 and then plus 10 to it so if you see this this equation or this particular formulation holds so 15 into 3 is 45 and 45 plus 10 is nothing but 55 let's take another example to illustrate it further so let us take another example to illustrate this so let me take a as 100 and b as let's say 12 okay and let us perform the long division method once again so how do we do it so this is 12 this is 100 okay and then I know 12 aids are 96 correct and hence I get a remainder four okay again if you see four is less than 12 here and hence we stop um let us take another such example maybe again if a is maybe let's say 200 and b let us take as 12 only okay so let us do this long division again so 12 and 200 and how do we do it so 12 ones are 12 so this is 12 and then you get eight as a remainder so you bring down zero right and then what 12 um 6 just 72 so if you see 12 6 72 so the remainder is 8 isn't it this is where now can and we proceed further no we can't proceed further why because what is the terminal or termination point so when you get 8 less than what 12 and then you stop in in every case for example here also you can express 100 as how can you express 100 100 can be expressed as 12 times 8 plus 4 you can check 12 aids are 96 plus 4 gives you 100 similarly here you can write 200 200 is equal to 12 times 16 plus 8 right so this is what Euclid actually proposed and now most generally we can we can you know uh say that so what was 100 in our previous case we learned that 100 was the dividend so dividend dividend is equal to what was 12 12 was divisor so dividend times divisor into what was 8 8 was nothing but quotient quotient quotient and then plus our remainder 4 in the above case this is what Euclid proposed now it may appear very obvious but during early days or let's say before the era where Euclid existed when Euclid existed this was you know very important result now in mathematical language we say that if we have if we have two two positive positive integers integers a and b a and b a and b then then there exists or there exist two positive integers two positive integers Q and R such that such that a equals b times Q plus R so any you you pick any two and we'll see in a little while you pick any two integers and then you'll always get a pair of integers Q and R such that a will be equal to b times Q plus R where where R R will be greater than equal to zero but less than equal to b so if you see remainder can be zero but remainder cannot be more than the quotient why because if the moment remainder goes beyond quotient so you can continue the division and you can get another remainder right which is lesser than b and we know that we stop dividing in the long division process when we get a remainder which is less than the divisor so this is the most common arithmetic tool or arithmetic phenomena which we have we have been observing over a period of time expressed in a mathematical language so this is what is called Euclid's division lemma so let me write this so this is what is called Euclid's Euclid's Euclid's division lemma now if you're not convinced maybe we can use few more examples to illustrate and let's take randomly any two integers so let me take another set of examples so let us take more examples example another example could be let's say you choose 15 as b and a you can take as let's say 78 okay let's randomly choose 15 and 78 so what will happen you will see 78 can be expressed as 15 times 5 which is 75 plus 3 so again in this case what is 78 78 was my a 15 was my b and we got q q as 5 and 3 as r another this thing you can have a little bit more let's say 25 is my b and a is let's say 423 okay this is another number so hence again if you try long division you can you can see that 423 423 can be expressed as 25 times 16 plus 23 so 25 times 16 is 400 and and if you add 23 to it you will get 423 so hence here again this is my a this is my b this is my quotient and this is my remainder again if you see 23 is this is less than my b which is 25 is it it now another example could be let's say 35 and 140 so if you see 140 is equal to 35 into 4 plus 0 so again if you see this is a this is b this one is c and this one r is 0 so r if you see r can be less than sorry greater than equal to 0 but less than equal to b this is what is meant by Euclid's division lemma this is used this is if you if you know later on in the succeeding successive sessions we'll see we'll we'll study about something called fundamental theorem of arithmetic and Euclid's division lemma is going to be very handy tool improving improving fundamental fundamental fundamental theorem of arithmetic we'll see that we'll see that in successive lectures right so hence understanding division Euclid's division lemma is very very important in the next session we'll try and prove Euclid's division lemma and what is meant by that it means that how do we know that if I choose any two integers a and b then what is the guarantee or what is the proof that every time I choose a random pair of integers positive integers I'll get another pair of positive integers which which satisfies this equation how do we know it it will happen always and we cannot just go on proving using some examples there could be a case where we get an example and it could fail so it's always better to have a generic proof and if that is true then and if we if we could do that then this is a well-established result that's what we will be looking to we'll be looking into in the next session thanks a lot for watching this 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