1976 AHSME Problem 19

A polynomial p(x) has remainder three when divided by x-1 and remainder five when divided by x-3. The remainder when p(x) is divided by (x-1)(x-3) is

\textbf{(A) }x-2\qquad \textbf{(B) }x+2\qquad \textbf{(C) }2\qquad \textbf{(D) }8\qquad \textbf{(E) }15

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