Use Euclid's division lemma to show that the cube of any positive integer is of the form 9m, 9m+1, or 9m+8.

Using Euclid division algorithm, we know that a=bq+r, 0≤r<b (1)
Let a be any positive integer, and b=3. Substitute b=3 in equation (1)
a=3q+r where 0≤r≤3, r=0,1,2
If r=0, a=3q
Cube the value, we get a³=27q³
a³=9(3q³), where m =3q³ (2)
If r=1, a=3q+1
Cube the value, we get a³=(3q+1)³
a³=(27q³+27q²+9q+1)
a³=9(3q³+3q²+q)+1, where m =3q³+3q²+q (3)
If r=2, a=3q+2
Cube the value, we get a³=(3q+2)³
a³=(27q³+54q²+36q+8)
a³=9(3q³+6q²+4q)+8, where m =3q³+6q²+4q (4)
From equation 2, 3 and 4,
The cube of any positive integer is of the form 9m, 9m+1 or 9m+8.