 Hello friends, welcome to another session of problem solving on finding the GCD using Euclid's division algorithm. In this question, we have to find the hcf of 72 and 56 and express it as a linear combination of 72 and 56. And also, we need to show that the linear combination is not unique. So if you remember what was GCD of, first of all, what are the requirements of this question? So first requirement is find GCD of 72 and 56. And second is to express GCD of 72 and 56 as linear combination of 72 and 56. So what was linear combination? Basically, we have to express GCD in terms of 72x plus 56y, where x and y are integers. And the third part is we have to show that x and y are not unique. What does this mean? That means we have to find out another set of x and y. Let's say x1 and y1 such that GCD of 72 and 56 should be equal to 72 times x1 plus 56 times y1. This is what we have to show. So there are three parts. So let us first find out the first part. For that, we have a table here which will be used to find out the GCD of 72 and 56 using what? We will be using the underlying concept here will be Euclid's division algorithm. So what was Euclid's division algorithm? So we start with A and B. So 72 is taken as A. So let's say 72 is A and my B is 56. And we divide 72 by 56. So if you see, 56 will go 1 times into 72 because 2 times 56 is approximately more than 100. 112 to be precise. So hence it will go 1 times. So hence 56 into 1 is 56 and the remainder in this case will be 16. So 56 plus 16 is 72. Now what happens? Next, the previous step B becomes A in the next step. So I have to write 56 here. So B comes here. This is equal to and R comes in place of B. So I'll write 16. Now we know 16 into 3 is 48. So it will go 3 times into 56. So hence if you see 48 plus 8 is 56. So 8 is the remainder. Now is the remainder 0? No. So you know that we continue this process till we get the remainder as 0. Now here again 16 will be put in A column. And here in B column we'll be putting the previous remainder that is 8. Now 8 into 2 is 16. So hence remainder is 0. So what is GCD? GCD is the last step divisor. So in the last step whatever is the divisor is the GCD according to Euclid's division algorithm. Okay. So 8 is my GCD. So what can I write here? I can write GCD of 72 comma 56 is equal to 8. Okay. So now let us go to the next part of the question. That means we have to express the GCD. So obtained 8 in terms of 72 and 56 as 72x and 56y. So let us attempt the second part of the question. So hence if you see from B there are three steps. So if you see I am enumerating them 1, 2 and 3. So from second step from step number 2 from 2 we can say 8 can be written as 56 into 1 minus 16 into 3. Right. So in the sidelines we can say that we have to express 8 as what now? 72x plus 56y. Is it it? So we have to attain the value of x. So this has to be obtained and this has to be obtained and then the second part is done. Now we can see 56 over here but you know we can't see 72 anywhere in this part. So what will I do? I will use the first step to express 16 as. So the same thing same 8 can be written as let 56 be as it is because we need 56. But 16 can be replaced by can you see this is nothing but 72 minus 56. In terms of 72 and 56 I have to reduce. So hence and this 3 comes down like that. Right. Now simplifying what we like it 8 is equal to 56 minus 72 into 3 plus 56 into 3. And further simplifying I can get 8 is equal to 56 into 4. Why? Because 1 and 3 4 and minus 72 into 3. So further rearrangement can show 72 into negative 3 plus 56 times 4. So if you see what is x value? So if you compare this one with this you will get x equals negative 3 and y equals 4. So this is the second part of the question. Now let us attempt the third part of the question. The third part says that show that the linear combination thus obtained is not unique. So what did we get? We get 8 equals 72 into minus 3 plus 56 into 4. So basically we have to find out another set of x and y that is x1 y1 such that 8 is equal to 72 into x1 plus 56 into y1. Can we do that? For such questions what do we do is this. So the same 8 can be represented as 72 into negative 3 plus 56 into 4 and you need to do what? Add and subtract two quantities so that the equation does not change. The value of the expression does not change. What do I add? I add 72 into 56 the same A and B and then to equate it back I subtract the same quantity again. And why do I do this? You will get to know just a little while later. So now take 72 common between first and third term. So what will you get? You will get 72 common and then this is minus 3 plus 56 minus 3 plus 56 and in the second term you will get 56 common and here it is 4 and minus 72 minus 72. Is it it? So hence finally you simplify what will you get? You will get 72 into 53 72 into 53 and here you will get 56 into negative 68 negative 68. So again if you see we are getting again a new set of x1 and y1. So if you compare these two these two so you will get what? x1 is clearly 53 and y1 is clearly negative 68. So we got another pair of integers x1 and y1. Previous pair was if you remember minus 3 and 4. x was minus 3 and y was 4. Here it is and now I am getting 53 and minus 68. That means we can get as many pairs as possible. So if you want another pair of x and y what do you need to do? Again add 72 into 56 and minus and subtract 72 into 56 from here and you will keep on getting more and more such pairs of x and y. That means the linear combination of GCD in terms of the two positive integers is not unique. That is what was the demand of the question and I hope you guys could understand this. Thanks a lot for watching this video.