If a prime number p

𝑝

 divides ab

𝑎

𝑏

, then which of the following is always true ?

Question 2Answer

a.

p/a

𝑝

/

𝑎


b.

p/b

𝑝

/

𝑏


c.

p/a

𝑝

/

𝑎

 or p/b

𝑝

/

𝑏


d.

p/(a+b)

The statement "If a prime number \( p \) divides \( ab \)," where \( a \) and \( b \) are integers, relates to a fundamental property of prime numbers. This property states that if a prime number \( p \) divides the product \( ab \), then \( p \) must divide at least one of the factors \( a \) or \( b \). This is known as **Euclid's lemma**.

### Therefore, the correct answer is:
**c. \( \frac{p}{a} \) or \( \frac{p}{b} \)**

### Explanation:
- **Euclid's lemma**: If \( p \) is a prime number and \( p \) divides the product \( ab \), then \( p \) must divide at least one of the factors \( a \) or \( b \).
- This means either \( p \mid a \) (i.e., \( \frac{p}{a} \) is an integer) or \( p \mid b \) (i.e., \( \frac{p}{b} \) is an integer).

### Additional Questions:
1. Can Euclid's lemma be extended to more than two factors? For example, if \( p \mid abc \), what can be concluded?
2. How does Euclid's lemma help in proving the uniqueness of prime factorization?
3. Can a non-prime number \( p \) satisfy the same condition as Euclid's lemma?
4. What is the significance of Euclid's lemma in number theory, particularly in the proof of the Fundamental Theorem of Arithmetic?
5. If \( p \) divides both \( a \) and \( b \), what can be said about \( p \)'s relationship with the greatest common divisor (GCD) of \( a \) and \( b \)?

### Tip:
Euclid's lemma is often used in proofs involving divisibility and is a key component in the proof that the square root of 2 is irrational.

Explore the concept of prime number division in number theory. Understand Euclid's lemma and its application to prime numbers dividing products of integers. Suitable for high school students learning number theory.

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