Let $\displaystyle n \in \mathbb{N}$. Are $\displaystyle n$ and $\displaystyle n+1$ relatively prime? How about $\displaystyle n$ and $\displaystyle n^2+1$?

My answer is yes, both $\displaystyle (n,n+1)=1$ and $\displaystyle (n, n^2+1)=1$. But is there a way to show these are true?