Question

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

Answer 1

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.