Question

Prove that if x and y are both odd positive integers, then x^2 +y^2 is even 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 + y2 is even but not divisible by 4.

Hope this helps you.

Cheers!

Answer 2

If  they both are Odd positive interger Then, it should be in form of (2n+1)
Say they have n value of n₁ and n₂
Then x=2n₁+1 and y=2n₂+1
x²+y²=(2n₁+1)² + (2n₂+1)²
4n₁² +4n₂²+4n₁ +4n₂ +1+1
2(2n₁² + 2n₂² + 2n₁ + 2n₂ +1)
Here the above is divisible by 2 but 2n₁² + 2n₂² + 2n₁ + 2n₂ +1 is not divisible by 2 and will give a remainder of 1 always.
So Hence,
if x and y are both odd positive integers, then x^2 +y^2 is even even but not divisible by 4.