Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.
Source: 2017 Romania JBMO TST 5.3
Tags: divides, number theory
Let $n \ge 2$ be a positive integer. Prove that the following assertions are equivalent: a) for all integer $x$ coprime with n the congruence $x^6 \equiv 1$ (mod $n$) hold, b) $n$ divides $504$.