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The problem statement in the image is as follows:

**Consider the set \(\mathbb{Z}\) of integers and an integer \(m > 1\). We say that \(a\) is congruent to \(b\) modulo \(m\), written \(a \equiv b \ (\text{mod} \ m)\), if \(m\) divides \(a - b\). Show that this defines an equivalence relation on \(\mathbb{Z}\).**

### Solution:

To prove that the relation \(a \equiv b \ (\text{mod} \ m)\) is an equivalence relation, we must show that it satisfies the three properties of an equivalence relation: reflexivity, symmetry, and transitivity.

1. **Reflexivity**:
   - For any integer \(a \in \mathbb{Z}\), we need to show that \(a \equiv a \ (\text{mod} \ m)\).
   - \(a \equiv a \ (\text{mod} \ m)\) means that \(m\) divides \(a - a\).
   - Since \(a - a = 0\), and \(m\) divides 0, this property holds true for any \(a\).
   - Therefore, the relation is reflexive.

2. **Symmetry**:
   - We need to show that if \(a \equiv b \ (\text{mod} \ m)\), then \(b \equiv a \ (\text{mod} \ m)\).
   - If \(a \equiv b \ (\text{mod} \ m)\), it means that \(m\) divides \(a - b\), i.e., \(a - b = km\) for some integer \(k\).
   - This can be rewritten as \(b - a = -km\), and since \(-k\) is also an integer, \(m\) divides \(b - a\).
   - Therefore, \(b \equiv a \ (\text{mod} \ m)\), so the relation is symmetric.

3. **Transitivity**:
   - We need to show that if \(a \equiv b \ (\text{mod} \ m)\) and \(b \equiv c \ (\text{mod} \ m)\), then \(a \equiv c \ (\text{mod} \ m)\).
   - If \(a \equiv b \ (\text{mod} \ m)\), then \(m\) divides \(a - b\), i.e., \(a - b = km\) for some integer \(k\).
   - If \(b \equiv c \ (\text{mod} \ m)\), then \(m\) also divides \(b - c\), i.e., \(b - c = lm\) for some integer \(l\).
   - Adding these two equations gives \( (a - b) + (b - c) = km + lm = (k + l)m \), which implies \(a - c = (k + l)m\).
   - Since \(k + l\) is an integer, \(m\) divides \(a - c\), and thus \(a \equiv c \ (\text{mod} \ m)\).
   - Therefore, the relation is transitive.

Since the relation \(a \equiv b \ (\text{mod} \ m)\) satisfies reflexivity, symmetry, and transitivity, it is an equivalence relation on the set \(\mathbb{Z}\).

### Summary:
The relation \(a \equiv b \ (\text{mod} \ m)\) is an equivalence relation on the set of integers \(\mathbb{Z}\) because it satisfies the three necessary properties: reflexivity, symmetry, and transitivity.

Would you like more details or have any questions?

### Related Questions:
1. What are some real-life applications of equivalence relations?
2. How does modular arithmetic play a role in computer science?
3. Can you provide an example of an equivalence class under this relation?
4. How does the concept of congruence modulo \(m\) relate to cyclic groups?
5. How is the Chinese Remainder Theorem related to modular arithmetic?

### Tip:
Remember that an equivalence relation partitions a set into disjoint equivalence classes. Understanding this concept is fundamental in various fields such as algebra and geometry.

Explore the concept of equivalence relations in modular arithmetic with a detailed explanation. Learn how the relation \(a \equiv b \ (\text{mod} \ m)\) defines an equivalence relation on the set of integers, including its properties: reflexivity, symmetry, and transitivity. Suitable for advanced high school students or college-level mathematics.

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