Exercise 1.2.53* ($n!+1$ and $(n+1)! +1$ are relatively prime.)

Let $d$ be an integer. Assume that $d\mid n!+1$ and $d\mid (n+1)!+1$. We must prove that $d\mid 1$.

From $n!\equiv -1(\mathit{mod}d)$, we deduce

This shows that $d\mid n$. Therefore $d\mid n!$, and $d\mid n!+1$, thus $d\mid (n!+1)-n!=1$.

This proves that $n!+1$ and $(n+1)!+1$ are relatively prime.