System of equations modulo p

Question

I've been working on a problem and I am stuck on a step in the proof, it might be trivial using linear algebra but I'm not sure. $p>10$ is a prime number and $a_{1} , a_{2} \dots, a_{9}$ are integers satisfying : $a_{1}^n +a_{2}^n+\dots a_{9}^n \equiv 0 \pmod p $ For infinitely many $n$. How can I show that $p$ divides at least one of $a_{i}$'s?

Answer 1

The statement as is is false. For a counterexample, take $p=19$, $a_1=a_2=a_3=1$, the rest of them are $3$, and $n$ to be congruent to $2$ mod $18$.