Question

prove that if x and y are both add positive integers, then x square + y square is even but not divisible by 4

Answer 1

I suggest writing them as 2α+1 and 2β+1, where α and β could be any natural number, odd or even. Then we could proceed by writing x² + y² as
   
(2α+1)² + (2β+1)² 
   
(4α² + 4α + 1) + (4β² + 4β +1)
   
4(α² + β² + α + β) + 2

So x² + y² is 2 more than a multiple of 4. Therefore it must be even, but can never be a multiple of 4 itself. See if you can prove this: if x and y are even numbers not divisible by 4, then x² + y² must be divisible by 8, but not 16.