Gcd(x^2+1,x+1) = {1 if x is even or 2 if x is odd?

Question

Divide x+1 into x^2+1 to get x-1 remainder 2. Divide 2 into x+1 to get x/2+1/2. Is this right? and How do I show it is even or odd? Thanks

Answer 1

If integer $d$ divides both,

$d$ will divide $x^2+1-(x-1)(x+1)=2$

If $x^2+1, x+1$ will be odd if $x$ is even, then $(x^2+1, x+1)=1$

What if $x+1$ is even?