Question

proove that diffrence of sauare of two odd natural numbers is a multiple of 8

Answer 1

Here are your two odd natural numbers: 
2m + 1 
2n + 1 

Square them, and take the difference. 
(2m+1)^2 - (2n+1)^2 
= 4(m^2 - n^2) + 4(m-n) 
= 4(m-n)(m + n + 1) 

Now just note that if m,n are both even or both odd, then m-n is even. If m,n has one even and the other odd, then m+n+1 is even.