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This problem requires proving the statement:  
\[
\forall n \in \mathbb{N} : (3 | n^2) \Rightarrow (3 | n)
\]
using the method of contraposition. The notation \(3 | n\) means "3 divides \(n\)."

### Solution Approach via Contraposition
The contrapositive of the implication \((3 | n^2) \Rightarrow (3 | n)\) is:
\[
\neg(3 | n) \Rightarrow \neg(3 | n^2)
\]
In other words, we need to show that if 3 does not divide \(n\), then 3 does not divide \(n^2\).

1. **Assume** \(3\) does not divide \(n\). This means that \(n \mod 3 \neq 0\).
   
2. **Possible Cases for \(n \mod 3\):**  
   Since \(n\) is not divisible by \(3\), there are two possibilities:
   - \(n \equiv 1 \pmod{3}\)
   - \(n \equiv 2 \pmod{3}\)

3. **Square Each Case:**  
   - If \(n \equiv 1 \pmod{3}\), then \(n^2 \equiv 1^2 = 1 \pmod{3}\).
   - If \(n \equiv 2 \pmod{3}\), then \(n^2 \equiv 2^2 = 4 \equiv 1 \pmod{3}\).

4. **Conclusion:**  
   In both cases, \(n^2 \equiv 1 \pmod{3}\), which means \(3\) does not divide \(n^2\).

Therefore, we have shown that if \(3\) does not divide \(n\), then \(3\) does not divide \(n^2\). By the contrapositive, this proves that if \(3\) divides \(n^2\), then \(3\) also divides \(n\).

This completes the proof.

Would you like further details, or do you have any questions?

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### Related Questions:
1. How would the proof change if we used direct proof instead of contraposition?
2. Can this method of contraposition be applied to similar divisibility problems?
3. How does modular arithmetic help simplify divisibility proofs?
4. Is there a different approach to show that \(3 | n^2\) implies \(3 | n\)?
5. What are some examples of using contraposition in other mathematical proofs?

### Tip:
When dealing with divisibility problems, modular arithmetic is often a helpful tool for simplifying the calculations and understanding the relationships between numbers.

Beweisen Sie die Aussage ∀n ∈ ℕ: (3 | n^2) ⇒ (3 | n) mittels Kontraposition.

Learn how to prove the divisibility statement ∀n ∈ ℕ: (3 | n^2) ⇒ (3 | n) using contraposition. This problem covers key number theory concepts, including modular arithmetic and divisibility, and demonstrates how contraposition is applied in mathematical proofs.

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