Consider the following statements in respect of two integers p and q (both > 1) which are relatively prime: 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.
1, 2 and 3
Understanding Relatively Prime Integers
The question asks us to consider three statements about two integers, p and q, both greater than 1, which are relatively prime. Let's break down what "relatively prime" means and then analyze each statement.
What are Relatively Prime Integers?
Two integers are said to be relatively prime (or coprime) if their greatest common divisor (GCD) is 1. This means they share no common positive divisors other than 1.
We are given that p > 1 and q > 1.
Analyzing the Statements
Let's examine each statement provided:
- Both p and q may be prime numbers.
A prime number is an integer greater than 1 that has no positive divisors other than 1 and itself. If p and q are both prime numbers, their only positive divisors are 1, p, and 1, q respectively.
- If p and q are different prime numbers (e.g., p=2, q=3), their only common positive divisor is 1. Thus, \(\text{GCD}(p, q) = 1\).
- If p and q were the same prime number (e.g., p=3, q=3), their GCD would be p (or q), which is greater than 1. However, the question implies considering scenarios where p and q are distinct or could be the same, but the condition for relative primality holds if they *may* be prime. Different primes are relatively prime.
Therefore, it is possible for both p and q to be prime numbers and be relatively prime. This statement is correct.
- Both p and q may be composite numbers.
A composite number is a positive integer greater than 1 that is not prime. It has at least one divisor other than 1 and itself. Can two composite numbers be relatively prime?
- Consider p = 4 and q = 9.
- Divisors of 4 are 1, 2, 4.
- Divisors of 9 are 1, 3, 9.
- The only common positive divisor is 1. \(\text{GCD}(4, 9) = 1\).
Here, both 4 and 9 are composite numbers (since 4 = 2 x 2 and 9 = 3 x 3), and they are relatively prime. This demonstrates that it is possible for both p and q to be composite numbers and be relatively prime.
Therefore, this statement is correct.
- One of p and q may be prime and the other composite.
Can a prime number and a composite number be relatively prime?
- Consider p = 2 (prime) and q = 9 (composite, 9 = 3 x 3).
- Divisors of 2 are 1, 2.
- Divisors of 9 are 1, 3, 9.
- The only common positive divisor is 1. \(\text{GCD}(2, 9) = 1\).
Here, one number is prime and the other is composite, and they are relatively prime. This occurs when the prime number is not a factor of the composite number.
It is also possible for them *not* to be relatively prime (e.g., p=2, q=4, \(\text{GCD}(2,4)=2\)). However, the statement says they *may* be, which is true as shown with p=2 and q=9.
Therefore, this statement is correct.
Conclusion on Statements
Based on our analysis with examples, all three statements correctly describe scenarios where two integers p and q (both > 1) can be relatively prime.
- Statement 1: Correct (e.g., p=2, q=3)
- Statement 2: Correct (e.g., p=4, q=9)
- Statement 3: Correct (e.g., p=2, q=9)
Thus, statements 1, 2, and 3 are all correct.
| Statement | Example (p, q) | p type | q type | GCD(p, q) | Relatively Prime? | Correct? |
|---|---|---|---|---|---|---|
| 1. Both prime | (2, 3) | Prime | Prime | 1 | Yes | Yes |
| 2. Both composite | (4, 9) | Composite | Composite | 1 | Yes | Yes |
| 3. One prime, one composite | (2, 9) | Prime | Composite | 1 | Yes | Yes |
Revision Table: Key Number Theory Concepts
| Term | Definition | Example |
|---|---|---|
| Integer | A whole number (positive, negative, or zero). | ..., -2, -1, 0, 1, 2, ... |
| Prime Number | An integer > 1 with only two positive divisors: 1 and itself. | 2, 3, 5, 7, 11, ... |
| Composite Number | An integer > 1 that is not prime (i.e., has more than two positive divisors). | 4, 6, 8, 9, 10, 12, ... |
| Greatest Common Divisor (GCD) | The largest positive integer that divides both numbers without leaving a remainder. | \(\text{GCD}(12, 18) = 6\) |
| Relatively Prime (Coprime) | Two integers whose GCD is 1. | \(\text{GCD}(7, 10) = 1\), so 7 and 10 are relatively prime. |
Additional Information on Relatively Prime Numbers
Relatively prime numbers are fundamental in number theory and cryptography. Here are some extra points:
- Two consecutive integers are always relatively prime. For any integer n, \(\text{GCD}(n, n+1) = 1\).
- If two numbers are relatively prime, they don't share any common prime factors. This is often the easiest way to check for relative primality. For example, to check if 4 and 9 are relatively prime, find their prime factorizations: \(4 = 2^2\), \(9 = 3^2\). They share no common prime factors (2 and 3), so their GCD is 1.
- If p is a prime number, then for any integer q, either \(\text{GCD}(p, q) = 1\) (if p does not divide q) or \(\text{GCD}(p, q) = p\) (if p divides q). So, a prime p is relatively prime to q if and only if q is not a multiple of p.
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