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

1.	Prove that if x and y are both odd positive integers then x2 + y2 is even but not divisible by 4.
Answer» Let us consider two odd positive numbers be x and y where x = 2p + 1 and y = 2q + 1 From question, x² + y² = (2p + 1)² +(2q + 1)² = 4p² + 4p + 1 + 4q² + 4q + 1 = 4p² + 4q² + 4p + 4q + 2 = 4 (p² + q² + p + q) + 2 Form above result, we can conclude that x and y are odd positive integer, then x² + y² is even but not divisible by four.

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

Answer»

Let us consider two odd positive numbers be x and y where

x = 2p + 1 and y = 2q + 1

From question,

x² + y² = (2p + 1)² +(2q + 1)²

= 4p² + 4p + 1 + 4q² + 4q + 1

= 4p² + 4q² + 4p + 4q + 2

= 4 (p² + q² + p + q) + 2

Form above result, we can conclude that x and y are odd positive integer, then x² + y² is even but not divisible by four.