Question 31
Let $p$ be a prime and let $a,b,c,n$ be positive integers with $a,b,c\lt p$ such that
$$p^2\mid a+(n-1)b,\qquad p^2\mid b+(n-1)c,\qquad p^2\mid c+(n-1)a.$$
Prove that $n$ is not prime.
$$p^2\mid a+(n-1)b,\qquad p^2\mid b+(n-1)c,\qquad p^2\mid c+(n-1)a.$$
Prove that $n$ is not prime.