please answer this question
Answers
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