Consider the following statements: 1) If p is relatively prime to each of q and r, then p is relatively prime to the product qr. 2) If p divides the product qr and if p divides q, then p must divide r.
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Let's analyze the given statements concerning number theory concepts like relative primality and divisibility.
Analyzing Statement 1: Relative Primality and Product
Statement 1 says: If p is relatively prime to each of q and r, then p is relatively prime to the product qr.
Two integers are relatively prime (or coprime) if their greatest common divisor (GCD) is 1. The statement can be written as:
If \( \gcd(p, q) = 1 \) and \( \gcd(p, r) = 1 \), then \( \gcd(p, qr) = 1 \).
Let's think about this. If p has no common prime factors with q, and p has no common prime factors with r, can p have any common prime factors with the product qr?
The prime factors of the product qr are simply the union of the prime factors of q and the prime factors of r.
Since p shares no prime factors with q (because \( \gcd(p, q) = 1 \)) and p shares no prime factors with r (because \( \gcd(p, r) = 1 \)), p cannot share any prime factors with the combination of prime factors of q and r, which form the prime factors of qr.
Therefore, p has no common prime factors with qr. This means their greatest common divisor is 1.
So, Statement 1 is correct.
Example:
- Let p = 5, q = 3, r = 7.
- \( \gcd(5, 3) = 1 \) (5 is relatively prime to 3).
- \( \gcd(5, 7) = 1 \) (5 is relatively prime to 7).
- The product qr = 3 * 7 = 21.
- \( \gcd(5, 21) = 1 \) (5 is relatively prime to 21).
- This example supports Statement 1.
Example where p is not prime:
- Let p = 6, q = 5, r = 7.
- \( \gcd(6, 5) = 1 \) (6 is relatively prime to 5).
- \( \gcd(6, 7) = 1 \) (6 is relatively prime to 7).
- The product qr = 5 * 7 = 35.
- \( \gcd(6, 35) = 1 \) (6 is relatively prime to 35).
- This example also supports Statement 1.
Analyzing Statement 2: Divisibility and Factors
Statement 2 says: If p divides the product qr and if p divides q, then p must divide r.
In mathematical notation:
If \( p | qr \) and \( p | q \), then \( p | r \).
The symbol \( p | qr \) means that qr is a multiple of p, or qr = kp for some integer k. Similarly, \( p | q \) means q = mp for some integer m.
Let's test this statement with an example. We need to find a case where \( p | qr \) and \( p | q \), but \( p \) does *not* divide \( r \).
- Let p = 6.
- Let q = 12. Here, \( p | q \) because 6 divides 12 (12 = 2 * 6).
- Let r = 5.
- The product qr = 12 * 5 = 60.
- Does \( p | qr \)? Does 6 divide 60? Yes, 60 = 10 * 6. So, \( 6 | 60 \).
- We have \( p | qr \) (6 | 60) and \( p | q \) (6 | 12).
- Now, does \( p | r \)? Does 6 divide 5? No.
In this example, we have \( p | qr \) and \( p | q \), but \( p \) does not divide \( r \). Therefore, Statement 2 is incorrect.
Statement 2 would be correct if p and q were relatively prime. This is a property known as Euclid's Lemma: If \( p | qr \) and \( \gcd(p, q) = 1 \), then \( p | r \). But Statement 2 does not include the condition that p and q must be relatively prime.
Summary of Statement Analysis
| Statement | Description | Correctness | Reason/Counterexample |
|---|---|---|---|
| Statement 1 | If \( \gcd(p, q) = 1 \) and \( \gcd(p, r) = 1 \), then \( \gcd(p, qr) = 1 \). | Correct | If p shares no prime factors with q and no prime factors with r, it shares no prime factors with qr. |
| Statement 2 | If \( p | qr \) and \( p | q \), then \( p | r \). | Incorrect | Counterexample: p=6, q=12, r=5. \( 6 | (12 \times 5) \) (6 | 60) and \( 6 | 12 \), but \( 6 \nmid 5 \). |
Based on our analysis, only Statement 1 is correct.
Revision Table: Key Number Theory Concepts
| Concept | Definition | Notation | Example |
|---|---|---|---|
| Divisibility | An integer a divides an integer b if b = ac for some integer c. | \( a | b \) | \( 3 | 12 \) because \( 12 = 3 \times 4 \) |
| Relatively Prime (Coprime) | Two integers a and b are relatively prime if their greatest common divisor (GCD) is 1. | \( \gcd(a, b) = 1 \) | \( \gcd(7, 10) = 1 \) |
| Greatest Common Divisor (GCD) | The largest positive integer that divides two or more integers without leaving a remainder. | \( \gcd(a, b) \) | \( \gcd(12, 18) = 6 \) |
Additional Information on Divisibility and Relative Primality
Understanding divisibility rules and properties of relatively prime numbers is fundamental in number theory. Statement 1 is a key property used in proofs involving coprime numbers.
Statement 2 highlights the importance of the relative primality condition. The implication \( \text{If } p | qr \text{ and } p | q \text{, then } p | r \) is generally false. It holds true if \( \gcd(p, q) = 1 \). In that specific case, any factor p shares with qr must come from r because p shares no factors with q (other than 1). Our counterexample (p=6, q=12, r=5) works precisely because \( \gcd(p, q) = \gcd(6, 12) = 6 \neq 1 \). Here, the factor 6 in the product qr (which is 60) comes partly from q (which is 12, a multiple of 6) and not necessarily from r.
These concepts are essential for topics like prime factorization, modular arithmetic, and cryptography.
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