Question

Show that the cube of any positive integer is in the form 9 and 9 m + 1 or 9m + 8 where m is some integer

Answer 1

9m, 9m+1, 9m+8
a=3q+r 0 less than equals to r less than 3
when r=0
a =3q
a cube=(3q) whole cube
= 9(q^3)
= 9m
where m is equal to q^3

when r=1
a=3q+1 ( by formula of (a+b) the whole cube
a^3= (3q+1) ^3
= 27q^3 + 27q^2+9q +1
=9(3q^3+3q^2+q) +1
=9m+1
where m is equal to 3q^3+3q+q

when r=2
a =3q+2
a^3= 27q^3+54q^2+ 18q+8
=9(3q^3+6q^2+2q) +8
=9m+8
where m is equal to 3q^3+ 6q^2+ 2q

hope this will help u