Question

show that square of any odd integer is of the Form 8 M + 1 where M is some integer

Answer 1

Any odd integer can be expressed in the form 2x+1. If you square it you get 4x²+4x+1.

4x²+4x= 4(x²+x). Whether x is even or odd, x²+x is divisible by 2. So 4(x²+x) is divisible by 4*2 or 8.

That mean that the square if any odd integer, 2x+1 is

4x²+4x+1 or (a multiple of 8)+1