Consider two integers p and q, both greater than 1, which are relatively prime (that is, gcd(p, q) = 1). Examine the following three statements: (1) Both p and q may be prime numbers. (2) Both p and q may be composite numbers. (3) One of p and q may be prime and the other composite. Which of the above statements are correct for such a pair (p, q)?

Introduction / Context:
This question explores the concept of relatively prime integers (also called coprime integers). Two integers p and q are relatively prime if their greatest common divisor is 1. The problem asks which combinations of “prime” and “composite” are possible for p and q when they are coprime and both are greater than 1.

Given Data / Assumptions:
p and q are integers, both greater than 1.p and q are relatively prime, meaning gcd(p, q) = 1.Statement (1): both p and q may be prime numbers.Statement (2): both p and q may be composite numbers.Statement (3): one of p and q may be prime and the other composite.

Concept / Approach:
To check whether each statement is possible, we do not need a general proof for all p and q; it is enough to exhibit one valid example for each statement that satisfies gcd(p, q) = 1. If we can find such examples, then the statements are correct. We will construct simple concrete pairs for each of the three cases.

Step-by-Step Solution:
Step 1: For statement (1), consider p = 2 and q = 3. Both are prime and gcd(2, 3) = 1, so statement (1) is possible.Step 2: For statement (2), consider p = 4 and q = 9. Both are composite (4 = 2 * 2 and 9 = 3 * 3), and gcd(4, 9) = 1, so both can be composite and still relatively prime.Step 3: For statement (3), consider p = 2 and q = 9. Here 2 is prime and 9 is composite, and gcd(2, 9) = 1, so one prime and one composite is also possible.Step 4: Because we have valid examples for each of the three statements, all three statements are correct.

Verification / Alternative check:
The only requirement for being relatively prime is that p and q share no common prime factor. This condition does not depend on whether p or q themselves are prime or composite. As long as their prime factorizations have no overlap, they are coprime. This general reasoning confirms that all three combinations are possible.

Why Other Options Are Wrong:
Any option that omits one of the statements (such as “1 and 2 only” or “1 and 3 only”) is incorrect because we have explicit examples showing that statements (1), (2) and (3) all occur. “2 and 3 only” wrongly excludes the very common case where both numbers are prime and coprime, such as 2 and 3.

Common Pitfalls:
A typical misconception is that if two numbers are relatively prime, both must be prime. This is false; many pairs of composite numbers, like 8 and 9 or 25 and 16, are relatively prime. Another misconception is to think that a prime number cannot be relatively prime with a composite number; again, 2 and 9 is a simple counterexample.

Final Answer:
All three situations are possible, so the correct choice is 1, 2 and 3.