1967 AHSME Problem 22

For natural numbers, when P is divided by D, the quotient is Q and the remainder is R. When Q is divided by D', the quotient is Q' and the remainder is R'. Then, when P is divided by DD', the remainder is:

\textbf{(A)}\ R+R'D\qquad \textbf{(B)}\ R'+RD\qquad \textbf{(C)}\ RR'\qquad \textbf{(D)}\ R\qquad \textbf{(E)}\ R'

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