 Hello and welcome to the session. In this session we will discuss about the real numbers. We will be discussing a very important property of positive integers in this session which is Euclid's division lemma. We know that a lemma basically is a proven statement which is used for proving another statement. Now let's state the Euclid's division lemma given positive integers a and b there exist unique integers q and r satisfying a is equal to bq plus r where r is greater than equal to 0 and less than b. Let's consider a pair of numbers a and b. Now we apply Euclid's division lemma to a and b. On doing this we get the question q and the remainder r where the remainder r is greater than equal to 0 and less than the number b. I hope you have understood how to apply Euclid's division lemma to given two positive integers. Next we discuss Euclid's division algorithm. This is based on Euclid's division lemma that we have just discussed. This is basically a technique to compute the highest common factor or you could say hcf of two given positive integers. This chart summarizes the Euclid's division algorithm. So if we need to find out the hcf of two numbers a and b then our first step would be to apply Euclid's division lemma to both the numbers a and b where a is taken to be greater than b to get the whole numbers q and r such that a is equal to bq plus r where r is given to be greater than equal to 0 and less than b. In the next step we check if r is equal to 0 or not if r comes out to be equal to 0 then the hcf of the numbers a and b would be equal to the number b and if r is not equal to 0 then we continue the step one that is we apply Euclid's division lemma to a and b where we take a as the value of b and b as the value of r and we continue this till we get r equal to 0. Let's try and find out hcf of two numbers a and b using Euclid's division algorithm. Here we have taken a to b greater than b now as in the first step we apply Euclid's division lemma to the numbers a and b to get the whole numbers q and r where r is greater than equal to 0 and less than the number b. Here as you can see that r is not equal to 0 then we repeat step one by taking a as the value of b and b as the value of r and now we apply the Euclid's division lemma to these two numbers. So we get the whole numbers q and r with r greater than equal to 0 and less than the number b. Again as we see that r is not equal to 0 so we take a as the value of b and b as the value of r and we apply Euclid's division lemma to this a and b. So we get the whole numbers q and r. Here as you can see that we get r equal to 0 so in this case we will stop and the hcf of the numbers a and b would be equal to the value of b at this step. Hence we have found out the hcf of two given numbers a and b using Euclid's division algorithm. So this completes the session. Hope you have understood the concept of Euclid's division lemma and Euclid's division algorithm.