1956 AHSME Problem 48

If p is a positive integer, then \frac {3p + 25}{2p - 5} can be a positive integer, if and only if p is:

\begin{array}{l} \textbf{(A)}\ \text{at least }3\qquad \textbf{(B)}\ \text{at least }3\text{ and no more than }35\qquad \\ \textbf{(C)}\ \text{no more than }35 \textbf{(D)}\ \text{equal to }35 \qquad \textbf{(E)}\ \text{equal to }3\text{ or }35 \end{array}

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