1975 AHSME Problem 22

If p and q are primes and x^2 - px + q = 0 has distinct positive integral roots, then which of the following statements are true?

\begin{array}{l} I.\ \text{The difference of the roots is odd.} \\ II.\ \text{At least one root is prime.} \\ III.\ p^2-q\ \text{is prime}. \\ IV.\ p+q\ \text{is prime} \end{array}

\textbf{(A)}\ \text{I only} \qquad \textbf{(B)}\ \text{II only} \qquad \textbf{(C)}\ \text{II and III only} \qquad \textbf{(D)}\ \text{I, II and IV only} \qquad \textbf{(E)}\ \text{All are true.}

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