67th NMO Selection Tests for JBMO

Let $n$ be an integer greater than $2$ and consider the set $A=\{2^n-1,3^n-1,\dots ,(n-1{)}^n-1\}$. Given that $n$ does not divide any element of $A$, prove that $n$ is a square-free number. Does it necessarily follow that $n$ is a prime number?

Marius Bocanu