Problem: If and are prime integers and has distinct positive integral roots, then which of the following statements are true?
I. The difference of the roots is odd.
II. At least one root is prime.
III. is prime.
IV. is prime.
Answer Choices:
A. I only
B. II and III only
C. All are true.
D. II only
E. I, II and IV only
Solution:
Since the product of the positive integral roots is the prime integer must be positive and the roots must be 1 and . Since is also prime, and . Hence all four statements are true.