Question

Show that square of any positive odd integer is in the form of 8q+1

Answer 1

since any +ve odd integer n is of the form 4q+1 or 4q+3
if n=4q+1
squaring both sides
n square=16q2+1+8q
              8(2q2+1q)+1
            8q+1 where q=2q+1
if n=4q+3
squaring both sides
n square=16q2+9+24q
              8(2q2+1+3q)+1
            8q+1 where q=2q2+1+3q