Calculate all prime numbers $x$, where $x^{18} - 1$ is divisible by $28728$

Question

Question: Calculate all prime numbers $x$, where $x^{18} - 1$ is divisible by $28728$

Apparently, the answer is all prime numbers except $2, 3, 7,$ and $19.$ I did some prime factorisation and found that $28728 =$ $2^3\times3^3\times7\times19$ and spotted that the numbers that $x$ cannot be with it. However, I still don't know why and how to prove that the answer can be all prime numbers except $2, 3, 7,$ and $19.$

Any help would be extremely appreciated :)

Answer 1

The Carmichael function $\lambda(n)$ gives the smallest exponent $k$, for which $$x^k\equiv 1\mod n$$ holds for all $x$ coprime to $n$.

Because of $\lambda(28728)=18$ , the congruence $$x^{18}\equiv 1\mod 28728$$ is satisfied for all $x$ coprime to $28728$ and in particular for all primes except $2,3,7$ and $19$.