Prove that if x and y are both odd positive integers then x2+y2 is even but not divisible by 4

Prove that if x and y are both odd positive integers then x2+y2 is eve

1.	Prove that if x and y are both odd positive integers then x2+y2 is even but not divisible by 4
Answer» Since x and y are odd positive integers, Then it should be in the form of 2x+1 (where x is a positive integer)Let x = 2m + 1 and y = 2n + 1 ( where m and n positive integers)Now x2 + y2= (2m + 1)2 + (2n + 1)2= 4m2 + 4m + 1 + 4n2 + 4n + 1= 4(m2 + n2 + m + n) + 2 ....... (1)from (1) we getx2 + y2 = 2 {m2 + n2 + m + n) + 1} = 2t (t = 2(m2 + n2 + m + n) + 1 is a positive integerTherefore it is clear that {tex}x^2+y^2{/tex} is an even number but not divisible by 4.