Question

Show that the square of any positive integer is of the form 3p,3p+1,3p+2

Answer 1

Suppose the positive integer is k.

Then By Division Algorithm,  
k=3q  or  k=3q+1  or  k=3q+2   where  q∈Z

CASE 1: 
k=3q
∴k²=(3q)²
∴k²=9q²
∴k²=3*3 q²
∴k²=3p         where p=3q²

CASE 2:

k=3q+1
∴k²=(3q+1)²
     =9q² + 6q + 1
     =3(3q²+2q) + 1
∴k²=3p+1       where  p=3q²+2q


CASE 3:

k=3q+2
∴k²=(3q+2)²
     =9q² + 12q + 4
     =9q² + 12q + 3 + 1
     =3(3q² + 4q + 1) + 1
∴k²=3p+1            where p=3q²+4q+1

Thus, square of any positive integer is of the form 3p or 3p+1.