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Let's go through each part of this problem, step-by-step.

### Given Information:
- Set \( A = \{1, 2, 3, 4, 5, 6\} \).
- Relation \( R \) on \( A \) is defined such that \( mRn \) if and only if \( 2 \mid (m + n) \), which means that \( m + n \) is even.

### Solution:
1. **(a) Directed Graph of \( R \)**:
   To construct the directed graph, we need to determine which pairs \( (m, n) \) satisfy the relation \( R \).
   
   - For two elements \( m, n \in A \), the sum \( m + n \) is even if both \( m \) and \( n \) are either both even or both odd.
   - Even numbers in \( A \): \( 2, 4, 6 \)
   - Odd numbers in \( A \): \( 1, 3, 5 \)
   
   Therefore:
   - \( mRn \) holds for pairs \( (m, n) \) where both \( m \) and \( n \) are either even or both are odd.

   **Pairs that satisfy \( R \):**
   - For odd elements \( \{1, 3, 5\} \): pairs are \( (1,1), (1,3), (1,5), (3,1), (3,3), (3,5), (5,1), (5,3), (5,5) \).
   - For even elements \( \{2, 4, 6\} \): pairs are \( (2,2), (2,4), (2,6), (4,2), (4,4), (4,6), (6,2), (6,4), (6,6) \).
   
   Using these pairs, you can create a directed graph with nodes labeled \( 1 \) to \( 6 \) and arrows connecting nodes according to the pairs above.

2. **(b) Equivalence Class \([5]_R\)**:
   The equivalence class \([5]_R\) consists of all elements \( n \in A \) such that \( 5Rn \).
   
   Since \( 5 \) is odd, we need to find all odd elements in \( A \). These are \( 1, 3, \) and \( 5 \).
   
   Thus, \([5]_R = \{1, 3, 5\}\).

3. **(c) Number of Distinct Equivalence Classes**:
   We observe that \( R \) groups elements into two classes based on parity:
   - Odd elements: \( \{1, 3, 5\} \)
   - Even elements: \( \{2, 4, 6\} \)
   
   These two groups represent two distinct equivalence classes under \( R \).

   Therefore, the relation \( R \) has **2 distinct equivalence classes**.

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Would you like further details on any part of the solution? 

### Related Questions
1. What makes a relation an equivalence relation, and how can you check it?
2. How do you interpret equivalence classes in terms of partitions of a set?
3. How would the solution change if \( A \) had a different number of elements?
4. Can you find equivalence classes in non-numeric sets, such as sets of words or other objects?
5. How would you generalize this relation for any finite set \( A \) with an arbitrary rule for even sums?

### Tip
To verify that a relation is an equivalence relation, always check that it is reflexive, symmetric, and transitive. These properties are essential for defining equivalence classes.

Let A = {1,2,3,4,5,6} and let R be a relation on A defined as follows: m R n ⟺ 2 | (m + n) for all m, n ∈ A. (a) Draw the directed graph of R. (b) Find the equivalence class [5]_R. (c) How many distinct equivalence classes does the relation R have?

Explore an equivalence relation defined on the set A = {1,2,3,4,5,6} where m R n if the sum of m and n is even. This problem covers directed graph representation, equivalence class [5]_R, and calculation of distinct equivalence classes. Ideal for Grades 10-12.

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