Question

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

Answer 1

I HOPE IT WILL HLP U .

   Any odd positive integer is of the form 2q + 1, where q is some integer.

   Let x = 2n + 1 and y = 2m + 1, where m and n are some integer.

   Now, x2 + y2 = (2n + 1)2 + (2m + 1)2

   => x2 + y2 = 4n2 + 4n + 1 + 4m2 + 4m + 1

   => x2 + y2 = 4(n2 + m2 + n + m) + 2

   => x2 + y2 = 4p + 2, where p = n2 + m2 + n + m

   => Since 4p and 2 are even numbers, So 4p + 2 is an even number.

   => x2 + y2 is an even number and leaves the remainder when divided by 4

   => So, x2 + y2 is an even but not divisible by 4