$29 \cdot 29$ When natural numbers $P$ and $P^{\prime}$ (where $P > P^{\prime}$) are divided by the natural number $D$, the remainders are $R$ and $R^{\prime}$, respectively. When $P P^{\prime}$ and $R R^{\prime}$ are divided by $D$, the remainders are $r$ and $r^{\prime}$, respectively. Then
(A) $r$ is always greater than $r^{\prime}$
(B) $r$ is always less than $r^{\prime}$
(C) $r$ is sometimes greater than $r^{\prime}$, sometimes less than $r^{\prime}$.
(D) $r$ is sometimes greater than $r^{\prime}$, sometimes equal to $r$
(E) $r$ is always equal to $r^{\prime}$
(20th American High School Mathematics Examination, 1969)
[Solution] Let $Q, Q^{\prime}, Q^{\prime \prime}$ be the quotients when $P, P^{\prime}, R R^{\prime}$ are divided by $D$, respectively. Thus,
$$
P=Q D+R, \quad P^{\prime}=Q^{\prime} D+R^{\prime}, \quad R R^{\prime}=Q^{\prime \prime} D+r^{\prime} .
$$

Therefore,
$$
\begin{aligned}
P P^{\prime} & =(Q D+R)\left(Q^{\prime} D+R^{\prime}\right) \\
& =\left(Q Q^{\prime} D+Q R^{\prime}+Q^{\prime} R\right) D+R R^{\prime} \\
& =\left(Q Q^{\prime} D+Q R^{\prime}+Q^{\prime} R+Q^{\prime \prime}\right) D+r^{\prime},
\end{aligned}
$$

Since $r^{\prime}<D$, the result of the division is unique.
Therefore, the remainder $r=r^{\prime}$ when $P P^{\prime}$ is divided by $D$.
Hence, the correct choice is $(E)$.