 Hello friends, welcome to the session I am Alka, let's discuss real numbers. Our given question is use Euclid's division lemma to show that the cube of any positive integer is of the form 9M, 9M plus 1, 9M plus 8. Now let X be any positive integer when it is of the form 3 cube plus 1, 3 cube plus 2. Now let's see the case first when X equal to 3 cube. Since I got into the question we have to show that the cube of any positive integer is of the form 9M or 9M plus 1, 9M plus 8. So when X equal to 3 cube on cubic both the sides we get X cube equal to 3 cube whole cube which is equal to 27 cube cube. This can also be written as 9 cube into 3 cube square. So we can say that X cube equal to 9M where equal to cube into 3 cube square. So here we have shown that the cube of any positive integer is of the form 9M. Now we see the case second when X equal to 3 cube plus 1. Now again on taking cube of both the sides we get X cube equal to 3 cube plus 1 whole cube. This is equal to 27 cube cube plus 27 cube square plus 9 cube plus 1 or we can say that X cube equal to on taking 9 cube common we get 3 cube square plus 3 cube plus 1 plus 1. This implies X cube equal to 9M plus 1 where M equal to cube into 3 cube square plus 3 cube plus 1 plus 1. So here again we have shown that the cube of any positive integer that is X is of the form 9M plus 1. Now we see the case third that is when equal to 3 cube plus 2. Now again taking cube of both the sides we get X cube equal to 3 cube plus 2 whole cube. This implies X cube equal to 27 cube cube plus 54 cube square plus 36 cube plus 8. This implies X cube equal to now we take 9 cube as common we get 3 cube square plus 6 cube plus 4 plus 8 or we can say that X cube equal to 9M plus 8 where M equal to cube into 3 cube square plus 6 cube plus 4. So here again we have shown that the cube of any positive integer is of the form 9M plus 8. Hence X cube is either the form 9M or 9M plus 1 or 9M plus 8. So hope you understood the solution and enjoyed the session goodbye and take care.