AHSME 1967 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: $\textbf{(A)}\ R+R'D \textbf{(B)}\ R'+RD \textbf{(C)}\ RR' \textbf{(D)}\ R \textbf{(E)}\ R'$