Question

If x and y are odd integers, then prove that x square + y square is even but not divisible by 4

Answer 1

So we know x and y are odd
let x = 2n+1, y = 2m+1
now
[tex] x^{2} + y^{2} = (2n+1)^{2} + (2m+1)^{2} \\ = (4n^2+4n+1) + (4m^2+4m+1) \\ = 4(n^2+m^2+n+m) + 2[/tex]
From above you can see that remainder from dividing the sum of squares of 2 odd numbers will be 2 always, hence its not divisible
And its even always