1999 AMC 12 Question 17

Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x - 19,$ the remainder is $99,$ and when $P(x)$ is divided by $x - 99,$ the remainder is $19.$ What is the remainder when $P(x)$ is divided by $(x - 19)(x - 99)?$