Question

Use division Algorithm to show that the square of any positive integer is of the form 3p3p+1,?

Answer 1

Let a be any positive integer
hee,b=3
r=0,1,2

r=0
a=bq+r
r=0
a=3q
a^2=(3q)^2 ( squaring both the sides)
3 (3q^2)=3m ( 3q^2=m)


r=1
a=3q+1
(a^2)=(3q+1)^2=9q^2+6q+1
3 (3q^2+2q)+1
3m+1 (m=3q^2+2q)

r=2
a=3q+2
a^2=(3q+2)^2=9q^2+12q+4
9q^2+12q+3+1
3 (3q^2+4q+1)+1
3m+1 (m=3q^2+4q+1)

instead of q you can write p
hope it will be correct