 Hello friends, so welcome to this new session on real numbers and Till now we were dealing with the Euclid's division lemma and in the last video we saw the geometrical interpretation of Euclid's division algorithm to find the greatest common divisor so if you remember we started with two strings of two different lengths and one was 42 another was of length 30 and then we tried to find out The method of dividing these two strings into segments which were of same length But the criteria was that the length should be highest length of the segment So formed should be highest and if you remember, this is what we did and we we we split it into two strings like this and Then we divided the the longer one in terms of the smaller one multiples of the smaller one and then we got 12 as leftover and finally We kept on repeating the process still we eventually landed up with seven strings of seven segments of length six each and Five of six each from the another one and then hence we declared that six was the greatest common divisor of 42 and 30 now today in this session we are going to study What exactly is greatest common divisor? So let's understand the term GCD or HCF as it's mentioned in many other books. So GCD stands for greatest greatest common Common Devices so we'll try and understand what does it mean and in your textbook you'd have also encountered another term called HCF and This is highest highest common Highest common Factor so you remember we discussed in the initial few sessions What was a factor? So hence the idea is to understand the meaning of the three words What words greatest common divisor or greatest common factor in reverse order that means we'll first take up What is divisor or a factor? So let us understand what is divisor? So we already know what is a divisor? So we say a divides B So this was a notation of a divides B And it means that it is what it means is we can find out a C such that B is equal to a Into C right so please remember a and B are we are integers Our integers so we say that a is a divisor of B when we can find out another integer C So if you see this is another integer another integer C such that A times C is B example So let's say I say 3 divides 12. Why because I can write 12 as 3 into 4 where 4 is another integer Now similarly 4 divides 32. Why because 32 is equal to 4 into 8 Yeah, so we say 4 divides 32 and 3 divides 12 right now We understand and so hence divisors are also called as factors and in many literature, especially in the Greek literature Early ancient Greek literature, you'll also see factors are also called measures. So they are also called Measure so you will see these terminologies in mathematical literature. They will either called GCT divisor or factor or measure so all these three words mean the same Okay, now, let's understand common the moment we are saying common that means we are dealing with more than one Integers so hence it can be a pair of integers. It could be three integers four so on and so forth So let's start with a pair of integers. So we say that We have let's say two two integers namely 60 and let's say we have 40 and We are interested in finding out all common common factors of common factors or devisors of of 60 and 40 Very easy method simple. Let's say the big number is not big So we don't need any algorithm. We can straight away find out The common divisor so let's say 60 if I have to enumerate the divisors of 60 You all know one is common to every integer So one common divisor of every integer so 60 divisors are one then we have two then we have three Then we have four four times 15 is 16 then five times 12 is 60 so five is also there seven is not there eight is not there nine is not there then comes ten Then comes ten then 11 not there 12. Yes, then 15 Then 20 then 30 and then 60 itself. So these are the These are the Factors or divisors of 60. Let's enumerate the divisors of 40. So one clearly is there then is two Then is four then five eight Then then again then 20 and then 40 Okay, so these are the the factors of 40 now the third step is to find out The common factors, so I hope I have written all the factors If I have not you guys are smart enough to figure out what all our common factors But just to re-check so it's one two three four five. Oh my god. Yes. I was wrong six is missing here So six and then seven eight Not there nine ten so six ten five twenty twelve fifteen four three twenty two thirty. Yeah, correct So this these are all now here five eight four ten He's not there two twenty one forty correct. So these are the factors of sixty and forty now Let us find out the Common factors. So one is common to both Two is also common to both. Okay, then four is also common to both. So these are common factors So that is that is what was meant by common factors. So five is also common to both Is it then again is a common factor? And in fact 20 should have been written here. So I'm writing here. So 20 is also a common factor. So if you see 20 is also common factor then None none other than these many so what all our common factors? Let us write down. So we are I'm writing common factors common factors factors of 40 and 60 There are a lot many how many in fact one two three four five six Six common factors are there. They are six in numbers six in numbers Now out of these six, I have to pick one which is the highest. Okay, so let's enumerate one two four five ten twenty So out of these which one is highest, you know This is highest. So hence this becomes my GCD This becomes my GCD. And how do I express this in mathematical literature? We say GCD of 40 comma 60 is equal to 20 Many literatures also would say simply 40 comma 60 within brackets is equal to 20 But to avoid confusion we usually mentioned or prefix GCD into this bracket. So we say GCD of A and B is equal to C where C is the highest common divisor of A and B Okay, this is how it is mentioned in mathematical literature Okay, now, let's take another example to see what GCD actually means or whatever. Let's say GCD of two numbers maybe Let's pick up two integers again and find out the GCD of these two numbers. So let us say the GCD of We are interested in finding out GCD of 72 and Let's say 20 These are A and B. So I'm interested in finding out GCD of 72 comma 20 is equal to what so let's quickly find out By the way, this is the way we are for trying to find out GCD here is not the way It is actually found out because if the numbers the integers are small you can always do that But what if the numbers are pretty large? Let's say you have to find out GCD of One zero two four and let's say four zero nine six in this case You cannot you know go by enumerating the factors and then finding the highest common factor No, we will talk about the different algorithms including Euclid's algorithm to find the GCD of two integers But here we this this particular practice or exercise is being done just to understand what GCD is Yeah, so once and for all you understand what GCD is. So now let us find out the factors of 72 So what all are the factors of 72? Let's enumerate again. So 72 will be or 72 factors are 1 then 2 then 3 then 4 and Then 5 is not there 6. Yes 6 is there 7 is not there 8 is there 9 is there Then 12 is there then 18 is there then 24 and then 36 and then finally 72 itself So how many 1 2 3 4 5 6 7 8 9 10 11 12 so there are 12 factors of 72 without the factors of 20 so if you see 20 is nothing but 1 2 3 is not there 4 then there is 5 6 is not there 7 8 not there 10 and Then finally 20 itself. So these are the factors of 20 now our objective is to find out common factors if you clearly see 1 is common to both 2 is common to both and then finally 4 is common to both apart from that. There are no common factors of 20 and 72 so hence my friends. What is the highest among 1 2 and 4 clearly? This one is GCD. So we write GCD or hcf of 72 comma 20 is 4 Right, so this is how GCD In this video what did we learn what is meant by greatest common divisor? Thank you