The task here is to prove the statement

$\forall n \in \mathbb{N}: \; (3 \mid n^2) \Rightarrow (3 \mid n)$

using the method of contrapositive. The statement $3 \mid n$ means "3 divides $n$."

Contrapositive Approach

The contrapositive of the implication $(3 \mid n^2) \Rightarrow (3 \mid n)$ is:

$\neg (3 \mid n) \Rightarrow \neg (3 \mid n^2)$

In words, this means: "If 3 does not divide $n$, then 3 does not divide $n^2$."

Proof of the Contrapositive

1. Assume $3 \nmid n$: This means that when $n$ is divided by 3, there is a remainder; $n$ is not divisible by 3.

2. **Analyze (n