AHSME 1975 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? $I.\ \text{The difference of the roots is odd.} \\ II.\ \text{At least one root is prime.} \\ III.\ p^2-q\ \text{is prime}. \\ IV.\ p+q\ \text{is prime}$ $\\ \textbf{(A)}\ I\ \text{only} \textbf{(B)}\ II\ \text{only} \textbf{(C)}\ II\ \text{and}\ III\ \text{only} \\ \textbf{(D)}\ I, II, \text{and}\ IV\ \text{only}\ \textbf{(E)}\ \text{All are true.}$