 Hey, hello friends. Welcome once again to another session on real numbers. So, let us continue with our process and so far we have discussed Euclid division algorithm and its application we saw in finding out its HCF or greatest common divisor that is GCD of any two positive integers. In this session let us try and understand one more concept and the concept is that GCD or HCF be of any positive integers A and B, A and B can be expressed as a linear combination. Please mind these words. So, linear combination of two integers, two integers x and y. So, hence it is written here that the GCD can be expressed as x times A plus y times B where x and y are integers but this representation is also not unique. So, you have to you have to you know check that also. So, though we can express any GCD of any two integers as xA plus yB but you can actually express in such linear combination in multiple ways. So, we will see what does this linear combination mean first of all and then we will try to also see why is this representation not unique. So, let us take an example as we have already shown you the method of finding out GCD and let us now try to use those examples to show that every such GCD found was or can be expressed as xA plus yB. Now, linear combination is whenever you express any integer in terms of another two or another set of integers such that the powers of the two integers here, can you see the power here is one. So, here also the power is one such representation is called linear combination because for any expression in which the variables are having power one, the graphical representation is linear. So, the linear word comes from there and combination is we are you know combining some factor or some multiples of A with some multiples of B to get deep hence linear combination of two integers. Let us take examples and see. So, if you remember we had two integers first was 42 and the second was 30. So, and GCD of this. So, we learned GCD of 42 and 30 which we had figured out was 6, isn't it. Now, 6 can be represented as some 6 can be represented as some 42 x plus I am sorry 6 can be represented as 42 x plus 30 y and the idea is to find the values of find the values of x and y. Right, let us see how. So, 6 is 42 x and 30 y. So, let us start with our our lemma only. So, if you remember Euclid's division lemma, 42 was expressed as 42 is equal to 30 times 1 plus 12 that is how we started finding GCD. So, hence and then the next step was 30 was equal to 12 into 2 plus plus 6 and then finally 12 was equal to 6 into 2. Okay. So, these are the three steps in which we found out the GCD of 30 and 42 to be 6. Now, let us let us do back calculation, backward calculation and see how we can express 6 as a linear combination of linear combination of 42 and 30. Now, let me say this was 1 and this was 2. So, from the second last step that is a equation number 2. What you can do is you can express 6 as, so 6 can be expressed as see 30 into 12, sorry 30 into 1 minus 12 into 2, isn't it? This is simple manipulation. So, 6 is 30 into 1 minus 12 into 2. Now, what is what was 12? What was 12? 12 was 12 can be written from here and same expression can be written as 30 into 1 minus 12 I can write as 42 minus 30 into 1 into 2. Why? Because from here if you see from here you can see 12, 12 is simply nothing but 42 minus 30 into 1, isn't it? Now, simplify what will you get? You will get 30 into 1 here the first term and then 42 into 42 into 2, 42 into 2 minus 30 into, sorry minus and minus will become plus here. So, minus minus plus 30 into 1 into 2 is 30 into 2. Now, what we do? We club these 30 together. So, we get what? 30 into 3 minus 42 into 2. Correct? Now, if you notice closely we could express 6 as 30 that was my b into y. So, this is y here. So, the order doesn't matter and then this is 42 and you can express this. I can write here as plus here and then shift this minus sign here. So, hence it becomes 42 into minus 2. Okay. So, what did we learn? We learned that 6 can be represented as 42 into minus 2 plus 30 into 3. So, this is a linear, this is called linear combination. So, if you saw, 6 was the GCD of 42 and 30 and I expressed 6 as linear combination, combination of of 42 and 30. Here, value of x was minus 2 and y is 3, isn't it? So, this was my expression into linear combination. Now, how do we know that it is not unique? We can take a simple example itself and prove that it is not unique. Let us do that here. So, if you see 6, the same expression I am writing here, 6 into 42 into minus 2 plus 30 into 3. Okay. This was my linear combination. Can I do this operation where I am adding 42 into 30 and subtracting 42 into 30? Okay. So, adding and subtracting would not change the equation, but this I did purposefully. Why? Now, let us rearrange this term and write like this. So, this is the same thing can be expressed as 42 into minus 2 plus 42 into 30 here and then plus 30 into 3 minus 42 into 30. Now, if you see, I can take 42 common in the first case. So, if you take 42 as common, you will get minus 2 plus 30, isn't it? And then second, you take 30 as common and it is 3 minus 42. And then simplifying, you will get 42 into 28, 28 plus or 30, 30 into minus 39. So, if you notice carefully, I have got another set of x and y. So, this is x here and this is let me just write both of them together. So, what did we found out? In the first case, I found out this was 42 into minus 2 plus 30 into 3 and the same 6 can be expressed as 42 into 28 plus 30 into minus 39. So, if you see guys, the same 6 have been expressed as same 6 has been expressed as 2 linear combinations. Okay. So, what is the conclusion? Conclusion is, conclusion is GCD of A and B is equal to x times A plus y times B where x and B are integers and x and y pair is not unique. We can get we can get many such pairs. This is what is meant by expressing the greatest common divisor of any two positive integers as a linear combination. So, this is my linear combination of the integers itself, linear combination. I hope you got the essence of this video and thanks a lot for watching it. Thank you.