Question

Use euclid's division Lemma to show that the cube of any positive integer is of the form :-
9m, 9m+1 or 9m+8

Answer 1

Let a be any positive integer
r=3q,3q,+1,3q+2
Case 1:a=3q
a=(3q)^3
a=27q^3
a=9(3q^3)
a=9m(were m=3q^3)
Case2:3q+1
a=(3q+1)^3
a=(3q)^3+(3q)^2(1)+3q(1)^2+(1)^3
a=27q^3+27q^2+3q+1
a=9(3q^3+3q^2+q)+1
a=9m+1(m=3q^3+3q^2+q)
Case3:3q+2
a=(3q+2)^3
a=(3q)^3+(3q)^2(2)+3q(2)^2+2^3
a=27q^3+54q^2+36q+8
a=9(3q^3+6q^2+4q)+8
Hence from above three cases we can conclude that the cube of any positive integer is of the form 9m,9m+1,9m+8

Answer 2

Hope it helps you .