After reading this question, I conjectured a generalization of it.
Conjecture: Fix $k\in \mathbb N$. Then, for all $n\in \mathbb N$, one of $n+1,\ldots,n+k$ is coprime to the rest.
I tried some elementary ways, but wasn't successful.
Observation: One of the consequences of this conjecture is that there are infinitely many primes!