prove 2 | n^2 + n + 1

Answer 1

if n is even, n^2+n+1 is odd
if n is odd, n^2 is odd, so n^2+n+1 is odd

I think you meant 2∤n^2+n+1

Answer 2

n^2+n+1 = (n+1)^2 - n
(n+1)^2 and n will always have the different parity, so the difference between them is always odd.