2.42 If $p$ and $q$ are prime numbers, and $x^{2}-p x+q=0$ has distinct positive integer roots, then which of the following statements is correct?
I. The difference of the roots is odd.
II. At least one root is prime.
III. $p^{2}-q$ is prime.
IV. $p+q$ is prime.
(A) Only I.
(B) Only II.
(C) Only II and III.
(D) Only I, II, and IV.
(E) All are correct.
(26th American High School Mathematics Examination, 1975)
[Solution] By Vieta's formulas, if the product of the two positive integer roots of the given equation is a prime number $q$, then the two roots must be 1 and $q$.
Since $p=1+q$ is also a prime, only $q=2, p=3$ satisfies this.
By $p^{2}-q=3^{2}-2=7$ being a prime, and $p+q=3+2=5$ also being a prime.
Therefore, conclusions I ~ IV are all correct.
Hence, the answer is $(E)$.