An integer `m` is said to be related to another integer `n` if `m` is a multiple of `n` . Check if the relation is symmetric, reflexive and transitive.

Explanation

To determine if the relation defined by "an integer \( m \) is related to another integer \( n \) if \( m \) is a multiple of \( n \)" is symmetric, reflexive, and transitive, we will analyze each property step by step.

### Step 1: Define the Relation
Let \( R \) be the relation defined as:
\[ R = \{ (m, n) \mid m, n \in \mathbb{Z} \text{ and } m \text{ is a multiple of } n \} \]
This means there exists some integer \( k \) such that:
\[ m = k \cdot n \]
where \( k \) is a natural number.

### Step 2: Check Reflexivity
A relation is reflexive if every element is related to itself. We need to check if \( m \) is related to \( m \):
- For any integer \( m \), we can write:
\[ m = 1 \cdot m \]
Here, \( k = 1 \) is a natural number.
- Since \( m \) is a multiple of itself, the relation is reflexive.

### Step 3: Check Symmetry
A relation is symmetric if whenever \( m \) is related to \( n \), then \( n \) is also related to \( m \). We need to check:
- Assume \( m \) is related to \( n \), which means:
\[ m = k \cdot n \]
for some natural number \( k \).
- To check for symmetry, we need to see if \( n \) is related to \( m \). This would mean:
\[ n = \frac{1}{k} \cdot m \]
- However, \( \frac{1}{k} \) is not a natural number (it is a rational number), which means \( n \) cannot be expressed as a multiple of \( m \) in the context of natural numbers.
- Therefore, the relation is **not symmetric**.

### Step 4: Check Transitivity
A relation is transitive if whenever \( m \) is related to \( n \) and \( n \) is related to \( o \), then \( m \) is also related to \( o \). We need to check:
- Let \( m \) be related to \( n \) and \( n \) be related to \( o \):
- \( m = k \cdot n \) for some natural number \( k \)
- \( n = l \cdot o \) for some natural number \( l \)
- Substituting the second equation into the first gives:
\[ m = k \cdot (l \cdot o) = (k \cdot l) \cdot o \]
- Since the product \( k \cdot l \) is also a natural number, \( m \) is a multiple of \( o \).
- Therefore, the relation is **transitive**.

### Conclusion
- The relation is **reflexive**.
- The relation is **not symmetric**.
- The relation is **transitive**.

### Summary
The relation defined by "an integer \( m \) is related to another integer \( n \) if \( m \) is a multiple of \( n \)" is reflexive and transitive but not symmetric.