algebraic proof

I think I can give it a go!

Proof : By the division theorem any number can be expressed in one of the forms 4q,4q+1,4q+2,4q+3

After squaring each of the odd expressions :

$(4q+1)^2= 16q^2+8q+1=8(2q^2+q)+1$

$(4q+3)^2=16q^2+24q+9=8(2q^2+3q+1)+1$

As shown here, it can be expressed as 8 times a number, then added by one. 

I hope this helps!