AHSME 1999 Problem 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 $(x-19)(x-99)$? $\mathrm{(A) \ } -x + 80 \mathrm{(B) \ } x + 80 \mathrm{(C) \ } -x + 118 \mathrm{(D) \ } x + 118 \mathrm{(E) \ } 0$