 Hello friends, so welcome again to get another session on real numbers today We are going to start a new topic and the topic name is Euclid's Euclid's division Euclid's division algorithm so We will get into the deeper details of Euclid's division algorithm in some while but before that I have a problem to solve here for all of you and This is a problem statement. The problem statement is I have Two strings one of let's say I name it string a and another string B One string is of 42 units long length and another string is of 30 units length Okay, now the question is I have to cut it into parts both the strings have to be cut into the part So criteria number one both the strings both the strings strings Have to be cut have to be cut Into parts into parts Constraint is a all The parts should be equal All the parts should be of equal length equal length Equal length second is All the parts would be of equal length and second is the length any length of each Each of the parts should be should be as Maximum as maximum As possible Isn't it? So what does it mean? This means that both the strings have to be cut into parts so okay, I can cut it into parts into let's say units of five six and ten and whatever and The concern but the constraint in country constraint is all the parts should be of equal length So after once the you know once I have cut both the strings When we look at the string lengths of each one of the substrings which you will get all Must be what of equal length first of all and second is the length of each of the parts Should be as maximum as possible. What does this mean? So for for that matter if you have a 42 unit long string a and 30 units long string B you can say that okay cut it into size of let's say one unit each so I'll get 42 units here 42 such pieces from string a and 30 such pieces in string B So what is what is that? What is what does that mean? It means that if you let's say cut one one part one one unit like that one one each and Then hence how many pieces will you get from here of one unit 42 and here you will get 30 So that satisfies this criteria number a that all the parts should be of equal length But can I is this the only possible way? Can I not cut it into you know? Let's say units of two. So let me say if I cut it into units of two Okay, then also if I cut it into units of two, I will get how many I will get definitely here how many 21 such Parts and in this case 15 such parts of this off unit two two units 15 two units part isn't it? So this is also this also satisfies all parts would be of equal length What is the length length of each of the parts two units each so 21 of two units each here and 15 of two units here, but again the question doesn't you know the process doesn't stop there So can I not cut it into let's say sizes of three each if I do that then what happens? I will get how many here 14 14 pieces of three units three units each and here I'll get 10 10 such pieces of three units each Isn't it so now the question is where do we stop so hence what should be the maximum can I have a let's say a side length of 40 each no, I can't have a segment of 40 units length. Why because if I cut a 40 I First of all, I can't cut the second string into a length of a size of 40. I can cut only the first one So that's not possible. So can I have let's say 25 as a size length No, I can't have that also why because if I cut it into 25 Here's Another string of or another segment of 17 units will be left over which is again, which wallets the criteria number one itself So what do I do? So for you know, so how do I do it? So do I keep on you know going for trial and error? So in this case you can easily see that as I reach six The case is solved, but then for larger pairs of numbers Let's say I have one not five and 95 or maybe some bigger number is there then. How do we do that for that? this guy Euclid the famous Greek mathematician of he was around 300 BC We have already learned that so this person this this man devised a method method And that's that method is called algorithm. So any method of let's say step-by-step Process of doing anything is called an algorithm So he devised an algorithm to solve this problem generally that means you have you may have some other numbers here Let's say 50 and 30 or 60 and 105 or whatever. So how to do that? That's what he this guy device. So what what did he do and this is the geometrical way of geometrical way of Doing this we will also see analytical or algebraic way of doing it and hence this guy Euclid is famous for his algorithm So he has mentioned this algorithm in his books called the elements and let's and let's understand what was the algorithm So he did this what did he do? So what he did was he took let's say What is the you know smaller one? So this 30 length is smaller one So I can carve out a 30 from the bigger one and what will be left over so if you see This will be left over right. So I'm just taking out the leftover once I have cut it into two pieces. So 130 and another this one Isn't it so this will be my first let's say Division so what I did I cut the bigger one and made it Let's say as many Units of the smaller one I could carve out so I could carve out only 30 and then I found out Here, sorry, this is 12. So if you see Yeah, so now the original 42 length and 30 length now are divided into three parts one of 30 another of 30 and Another leftover is 12, but it doesn't satisfy our criteria a which one this one All the parts should be of equal length. So our is 12 equal to 30 No, that means I love to go for a further division. So hence now what I'll do I will carve out. I will try to, you know, cut the bigger remaining strings into pieces of length 12. So let's see how Okay, so here is that two strings, which I have now, you know, three strings basically so the 42 length String was cut into 30 and 12 and the 30 was left as it is now I will cut both these longer strings into Strings of length 12. So let me take out the first 12 from this string So I am taking this out and hence I am putting it here. Okay, so this is gone. So if you see This is 12 length. Can I take out one more 12 from here? Yes, I can so because it is of 18 length So here is how I take out one more and Sorry, so this is This is how I do it. I take out one more and put it here Okay, now can I take 12 further from this? No, not possible. So let me do the same exercise here also So I took out 12 from here and put it side by side. Okay, and this becomes my 12 length. Can I take one more 12? Yes, I can So I took out one more 12 from here and put it here So now can I do anything further? No, I can't. So hence I divided the original 42 into 12, 12, 12, 12 and 6 and the 30 as 12, 12 and 6 Now does it satisfy the initial criteria? The first one all the parts would be of equal length still not why because few are of 12, few are of 6. Then what do we do? Then again, we'll repeat the process. Now what we'll do is we'll carve out 6 from strings of length 6 from all of these remaining. How many are there? 1, 2, 3, 4, 5, 6, 7. So we have 7 such You know strings of 5 of them are of 12 and 2 are of them 6. So this is the case now What do we do? Let me extract out 6 from each one of them. So I can take out the 6 So if you see I am putting it here. Okay, so Now this is of 6. Okay. Very good. Now. What what about this? I can take out here also 6 Let me put it here. So this is also of 6 and then I can take out this 6 as well and put it here So this is of 6 also. So if you see all are now 6 length now do the same exercise here and This is 6 Okay, if you see this is 6 and then This also if you take out here. Yeah, so this also is 6 So if you see we have now all the the entire original 42 length string was cut into 7 pieces of 6 each and The 30 length string was cut into 5 pieces of 6 each. Does it satisfy a? Yes, all the parts should be of equal length. Yes, we have got all the parts of equal length now here You cannot really have any other You know size of the string which can divide the two strings in equal length So hence this is what is called the greatest common factor or greatest common divisor of two numbers 42 and 42 and 30 so if you see 6 becomes the 6 is the Greatest great test is Greatest is the thing which we are trying to find out As the you know the second criteria be it has to be the greatest one and it is common common Common and third is Devise it so Euclid's division algorithm is used to find out greatest common divisor of greatest common divisor of two integers two positive integers two integers and positive is in it and You know mentioned here. So there are two integers and in this case the integers were 42 and 30 Okay, so we will look into the deeper details of how GCD is found out using Euclid's division algorithm because we can't draw images and You know representative diagrams to find this out. This was just to explain what it practically means How do we segment? Let's say our two things into equal sizes to two objects into equal sizes such that The sizes are first of all All are equal and second is they are maximum So I hope you understood the process you can do it You know Yourself you can take a sheet of paper and make strips of let's say some unit some integral size and then Try and find out the greatest common divisor using this Algorithm. Thank you for watching this video