Question

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

Answer 1

Solution:
Since x and y are odd positive integers, we have
x = 2m + 1 and y = 2n + 1
=> x² + y² = (2m + 1)² + (2n + 1)²
= 4m² + 4m + 1 + 4n² + 4n + 1
= 4(m² + n²) + 4(m + n) + 2
Hence, x² + y² is an even number but not divisible by 4.