The task asks to indicate all the correct statements related to equivalence relations. Let’s go through each statement one by one:

1. Statement 1:
   Let $X$ be the set of all natural numbers, and $m \sim n$ if and only if $m$ and $n$ are coprime. This relation is an equivalence relation.
   This statement is incorrect. The property "coprime" means that two numbers share no common divisors other than 1, but this does not satisfy the transitivity condition required for an equivalence relation. Transitivity would mean that if $m \sim n$ and $n \sim p$, then $m \sim p$, which is not necessarily true for coprime numbers.

2. Statement 2:
   The relation of congruence modulo 2 divides the set $\mathbb{Z}$ into three equivalence classes.
   This statement is incorrect. Congruence modulo 2 divides the set of integers $\mathbb{Z}$ into two equivalence classes: even numbers and odd numbers.

3. Statement 3:
   Let $X$ be the set of all triangles, and $m \sim n$ if and only if both $m$ and $n$ are acute triangles. This relation is an equivalence relation.
   This statement is correct. The relation "both triangles are acute" satisfies the conditions of an equivalence relation: reflexivity, symmetry, and transitivity.

4. Statement 4:
   If a set $X$ has an equivalence relation, it induces a partition of this set into equivalence classes.
   This statement is correct. By definition, an equivalence relation on a set induces a partition of the set into disjoint equivalence classes, where elements in the same class are equivalent to each other.

Conclusion:

• Correct statements: 3 and 4.

Let me know if you need further clarification or details.

Here are 5 related questions for further exploration:

1. What are the key properties of an equivalence relation?
2. How does the concept of partitioning a set relate to equivalence relations?
3. Can you provide an example of a relation that is symmetric and reflexive but not transitive?
4. How does modular arithmetic create equivalence classes in number theory?
5. What are the implications of a relation being transitive but not reflexive?

Tip: Always check for the three properties—reflexivity, symmetry, and transitivity—when determining if a relation is an equivalence relation.