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To prove the statement that if \( n \) is composite, then there exists a prime \( p \leq \sqrt{n} \) such that \( p \mid n \), we will follow these steps:

1. **Definition and Assumptions**: 
   - A composite number \( n \) is a number greater than 1 that is not prime; in other words, it has divisors other than 1 and itself.
   - We need to show that there exists a prime \( p \leq \sqrt{n} \) that divides \( n \).

2. **Consider the Properties of Composite Numbers**:
   - If \( n \) is composite, it can be expressed as the product of two integers greater than 1, say \( a \) and \( b \), such that \( n = a \cdot b \).

3. **Using the Hint**:
   - Consider the scenario where both \( a \) and \( b \) are greater than \( \sqrt{n} \). If \( a > \sqrt{n} \) and \( b > \sqrt{n} \), then their product \( a \cdot b \) would be greater than \( \sqrt{n} \cdot \sqrt{n} = n \), which contradicts the fact that \( n = a \cdot b \).

4. **Conclusion from the Hint**:
   - Since \( a \) and \( b \) cannot both be greater than \( \sqrt{n} \), at least one of them must be less than or equal to \( \sqrt{n} \). Without loss of generality, assume \( a \leq \sqrt{n} \).

5. **Prime Factorization**:
   - Now, if \( a \) is not prime, then \( a \) can be factored further into prime factors all of which must also be \( \leq \sqrt{a} \leq \sqrt{n} \).

6. **Existence of Prime \( p \leq \sqrt{n} \)**:
   - By the fundamental theorem of arithmetic, every integer greater than 1 can be factored uniquely into prime numbers. Therefore, the number \( a \) (or \( b \)) can be factored into prime numbers, and at least one of these primes must be \( \leq \sqrt{n} \).

7. **Final Statement**:
   - Thus, there exists at least one prime \( p \) such that \( p \leq \sqrt{n} \) and \( p \mid n \), completing the proof.

Would you like a more detailed breakdown of any specific step? Here are some questions you might find useful to ask:

1. Can you show an example with specific numbers to illustrate this proof?
2. How does the proof change if \( n \) is prime?
3. What are the implications of this proof for prime factorization algorithms?
4. Can you explain the significance of the square root in this context?
5. How would you use this proof in a practical application?

**Tip**: When tackling mathematical proofs, break down the problem into smaller logical steps and consider any provided hints to guide your reasoning.

Learn how to prove the statement that if n is composite, there exists a prime p ≤ √n such that p divides n. This problem delves into number theory concepts, including prime numbers, composite numbers, and prime factorization. Suitable for advanced high school students and beyond.

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