Question

prove that the square of any positive integer is of the form 4 Q are 4 Q + 1 for some integer q

Answer 1

let a be any positive integer and b=2
by Euclid division lemma
a=bq+r where 0<r<=b
1) r=0
a= 4q+0
a=4q
squaring on both side
a^2=(2q)^2
a^2=4q
2) r=1
a= 2q +1
squaring on both side
a^2 =(2q+1)^2
a^2=4q+1+4q
a^2= 4q(q)+1
a^2=4q*integer+1
a^2=4q+1
then the square of any positive integer is of the form 4q and 4q+1