Why are $x$, $x^3+1$ and $x^2+x+1$ always mutually co-prime, for any natural number $x$? I just read that for a natural number $x$, the three numbers $x$, $x^3 + 1$ and $x^2 + x + 1$ are all mutually co-prime.
I couldn't find a reason why this is true. OK, any of them does not divide either of the other two, but is it enough to conclude that their GCD is 1?
Couldn't they have any common factors?
Also on what basis, are two consecutive integers co-prime? I know they cannot have a common factor but don't know why. 

If integer $d$ divides both $x^2+x+1, x^3+1;$
$d$ must divide $x(x^2+x+1)-(x^3+1)=x^2+x-1$
$\implies d$ must divide $x^2+x+1-(x^2+x-1)=2$
But $2\mid x(x+1)\implies x^2+x\pm1=x(x+1)\pm1$ are odd $\implies d=1$
