This statement is true and can be justified using the fundamental properties of prime numbers and divisibility in integers.

Proof:

1. Given: $p$ is a prime number, $m, n \in \mathbb{Z}$, and $p \mid (mn)$.
   This means that $mn$ is divisible by $p$, i.e., there exists some $k \in \mathbb{Z}$ such that
   $mn = pk$

2. Since $p$ is prime, one of its key properties is that it cannot divide a product without dividing at least one of its factors (this is a consequence of the Fundamental Theorem of Arithmetic).

3. Using Euclid's Lemma:

   • If $p$ divides a product $mn$, then $p$ must divide either $m$ or $n$, or both.

4. Conclusion:
   If $p \mid mn$, then $p \mid m$ or $p \mid n$, where $p$ is a prime number.

This property holds because prime numbers have no divisors other than 1 and themselves, making them "irreducible" in the sense of divisibility. If $p$ did not divide either $m$ or $n$, their product $mn$ could not be divisible by $p$, contradicting the initial assumption.

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Would you like further clarification or examples of this principle?
Here are 5 related questions to explore further:

1. Can this property extend to composite numbers $p$? Why or why not?
2. How does this relate to the Fundamental Theorem of Arithmetic?
3. What happens if $p \mid (m + n)$ instead of $mn$?
4. Can we generalize this property to non-integers, such as in the set of rational numbers $\mathbb{Q}$?
5. What are some practical applications of this property in number theory?

Tip: Understanding Euclid's Lemma is crucial in modular arithmetic and cryptography, as it underpins the behavior of prime numbers in mathematical systems.