Question

please answer this question ​

Answer 1

Answer:

Step-by-step explanation:

according to euclids algoritham a=bq+r , 0<r<b

we have to prove cube of any positive integers is in the form of

9m,9m+1,9m+8

let a be any positive integer and b=9

the positive integers are 0,1,2,3,4,5,6,7,8

a = 9q+0,9q+1,9q+2,9q+3,9q+4,9q+5,9q+6,9q+7,9q+8

cube of any positive integer

(9q+0)^3

            =(9q)^3

             =729q^3

               =9(81q^3)

                  =9m

(9q+1)^3

  (a+b)^3= a^3+b^3+3ab(a+b)

(9q+1)^3= (9q)^3+(1)^3+3(9q)(1)(9q+1)

              =729q^ +1 27q(9q+1)

              =  729q^3+243q^2+27q+1

              =9(81q^3 +27q^2+ 32)+1÷m

               =9m+1

(9q+8)^3 = (92)^3+8^3+3(9q)8(9q+8)

              =729q^3+512+216q(9q+8)

             =729^3+1944q^2+1728q

           = 729q^3+1944q^2+1728q+512)^2

          =9(81q^3+216q^2+192q+56)+8÷m

           =9m+8

therefour cube of any positive integer