What is the gcd of x+1 and x-1?

Question

What is the $gcd$ of $x+1$ and $x-1$? The euclid algorithm says that it is 2, but I'm unsure, since if I divide $x+1$ by $x-1$, the remainder is 2.

Answer 1

Rather, the Euclidean algorithm shows in this way that their greatest common divisor is at most $2.$ Can you figure out for which $x$ it is equal to $2$ and for which it is equal to $1$?