As soon as I saw the word "Which...", I knew that there's a list of choices
to go along with this question, and I was pretty sure that when I opened
the question, the list wouldn't be there. Thanks for not disappointing me.
One integer can be 'x' . Then the next consecutive integer is (x+1).
Let's look and see what the sum of their squares looks like:
(x)² = x²
(x+1)² = x² + 2x + 1
Their sum: (x²) + (x² + 2x + 1) = 2x² + 2x + 1 .
That's the expression we need. Look at it closely.
No matter what 'x' is ...
-- (2x²) is definitely an even number
and
-- (2x) is definitely an even number
so
-- (2x² + 2x) is definitely an even number
and
-- (2x² + 2x + 1) (the sum of the integers) is definitely an odd number.
QED