1999 AHSME 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 \qquad \mathrm{(B) \ } x + 80 \qquad \mathrm{(C) \ } -x + 118 \qquad \mathrm{(D) \ } x + 118 \qquad \mathrm{(E) \ } 0

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.