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.