Question

Prove that if x and y are both odd positive integers then x^2+y^2 is even but not divisible by 4.

Answer 1

since x and y are odd numbers,

Let x = 2p+1

y = 2q + 1

x² + y² = (2p+1)² + (2q+1)²

=> 4p² + 4q² + 4p + 4q + 2

above eq. can be written as

1) => 2 × (2p² + 2q² + 2p + 2q + 1)   => Even number (multiple of 2)

or 2)  4×(p² + q² + p + q) + 2   => Not divisible by 4 (remainder is 2 after division by 4)

Hence proved