MathNet 2013 Question 88

Let $a_{1}, a_{2}, \ldots, a_{9}$ be integers. Prove that if $19$ divides $a_{1}^{9} + a_{2}^{9} + \cdots + a_{9}^{9}$ then $19$ divides the product $a_{1} a_{2} \cdots a_{9}$.