Question

prove that square of any positive integer is 4q or 4q+1 for some integers q

Answer 1

let x be any positive integer and b=4 , so i.t is of the form 4q,4q+1 now refer to pic

Answer 2

by euclid's division Lemma, we know,
a=bm+r
where b=2, r=0,1 ( as b>r>or=0) and a is any positive integer

case 1
b=2, r=0
a=2m
squaring both sides
a^2= 4m^2 ( where q =m^2 is some integer)
a^2=4q

case 2
b=2,r=1
a=2m+1
squaring both sides
a^2=4m^2+4m+1
a^2=4m(m+1)+1 ( where q=m(m+1) is some int.)
a^2=4q+1
hence proved that the square of any positive integer is 4q or 4q + 1 for some integer q