$29 \cdot 64$ If $p$ is a positive integer, then $\frac{3 p+25}{2 p-5}$ is a positive integer if and only if $p$ is
(A) not less than 3.
(B) not less than 3 and not greater than 35.
(C) not greater than 35.
(D) equal to 35.
(E) equal to 3 or 35.
(7th American High School Mathematics Examination, 1956)
[Solution] Let $\frac{3 p+25}{2 p-5}=n, n$ be a positive integer.
Obviously, if $2 p-5=k$, then $3 p+25=n k$.
Eliminating $p$ gives $k(2 n-3)=65=1 \cdot 65=5 \cdot 13$, so
$$
\left\{\begin{array}{l}
k=1,65,5,13 ; \\
2 n-3=65,1,13,5 .
\end{array}\right.
$$

Then $p=3,35,5,9$.
Therefore, the answer is $(B)$.