We want to show that if $n^2$ is a multiple of 3, then $n$ must also be a multiple of 3. We'll use a proof by contrapositive and modular arithmetic.

Proof by Contrapositive:

We will prove the contrapositive statement:
"If $n$ is not a multiple of 3, then $n^2$ is not a multiple of 3."

The contrapositive is logically equivalent to the original statement. So, if we can prove this, the original statement will be true.

Step 1: Modular Arithmetic and Cases

If $n$ is not a multiple of 3, then $n$ can either be congruent to 1 or 2 modulo 3. In other words, we consider the following cases: $n \equiv 1 \ (\text{mod} \ 3) \quad \text{or} \quad n \equiv 2 \ (\text{mod} \ 3)$

We will square each case and see if $n^2$ can be divisible by 3.

Case 1: $n \equiv 1 \ (\text{mod} \ 3)$

$n^2 \equiv 1^2 \equiv 1 \ (\text{mod} \ 3)$ So, if $n \equiv 1 \ (\text{mod} \ 3)$, then $n^2 \equiv 1 \ (\text{mod} \ 3)$, which means $n^2$ is not divisible by 3.

Case 2: $n \equiv 2 \ (\text{mod} \ 3)$

$n^2 \equiv 2^2 \equiv 4 \equiv 1 \ (\text{mod} \ 3)$ So, if $n \equiv 2 \ (\text{mod} \ 3)$, then $n^2 \equiv 1 \ (\text{mod} \ 3)$, which again means $n^2$ is not divisible by 3.

Step 2: Conclusion

In both cases, when $n$ is not a multiple of 3, $n^2$ is also not divisible by 3. Hence, the contrapositive is true, and therefore, the original statement is also true:

$\text{If } n^2 \text{ is a multiple of 3, then } n \text{ must be a multiple of 3.}$

This completes the proof.

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Would you like a deeper explanation of any part? Here are 5 related questions to explore:

1. How does modular arithmetic work for different numbers, not just 3?
2. Can we generalize this result for other prime numbers like 5 or 7?
3. What other proofs can be made using the contrapositive method?
4. How does the original assumption help us solve problems in number theory?
5. Can this proof method work for non-integer values like rational numbers?

Tip: When tackling divisibility problems, modular arithmetic is a powerful tool for simplifying and solving congruence relations.