Question

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

Answer 1

Let the two odd positive numbers be x = 2k + 1 a nd y = 2p + 1
Hence x² + y² = (2k + 1)² + (2p + 1)²
= 4k² + 4k + 1 + 4p² + 4p + 1
= 4k² + 4p² + 4k + 4p + 2
= 4(k²+ p² + k + p) + 2
Clearly notice that the sum of square is even the number is not divisible by 4.
Hence if x and y are odd positive integers, then x² + y²is even but not divisible by 4