I need to prove that
$$ \gcd(n!+1,(n+1)!+1) = 1 $$
In class, the teacher advised us to use Bézout's Identity. So I wrote the following
$$ x(n!+1) + y[(n+1)! + 1] = 1 $$
But from there I'm at a loss.
To make the question more generic: I don't understand how Bézout's Identity can help me prove that the $\gcd$ of two numbers is really $n$.
Previous exercises I had to prove were much simpler; for example, I had to prove that $\gcd(2n+1, 3n+1) = 1$. In this case, I just tried to find $x$ and $y$ that make the statement $x(2n+1) + y(3n+1) = 1$ true. And by finding any two integers, I understood that I had provided enough proof. But I didn't do any math to find $x$ and $y$ in that simpler case... or I did, but I don't understand what I did... anyway, I could just "see", right away, that if $x=3$ and $y=-2$, then it would work out.
Can you help me understand how to use Bézout's Identity in proofs of any type? I think I'm not using it correctly.