State and prove congruence modulo m is an equivalence relation.

**Statement:**

Congruence modulo \( m \) is an equivalence relation.

**Definition:**
For any integer \( m \geq 1 \), we say that two integers \( a \) and \( b \) are congruent modulo \( m \), denoted by \( a \equiv b \pmod{m} \), if \( m \) divides \( a - b \). In other words, \( a \equiv b \pmod{m} \) if there exists an integer \( k \) such that:
\[
a - b = km
\]

To prove that congruence modulo \( m \) is an equivalence relation, we need to show that it satisfies the three properties of an equivalence relation: **reflexivity**, **symmetry**, and **transitivity**.

### 1. Reflexivity:
We need to show that for any integer \( a \), \( a \equiv a \pmod{m} \).

**Proof:**
Consider \( a - a = 0 \). Since \( 0 \) is divisible by \( m \) (i.e., \( 0 = 0 \times m \)), it follows that \( a \equiv a \pmod{m} \). 

Thus, the reflexivity property holds.

### 2. Symmetry:
We need to show that if \( a \equiv b \pmod{m} \), then \( b \equiv a \pmod{m} \).

**Proof:**
If \( a \equiv b \pmod{m} \), by definition, \( a - b = km \) for some integer \( k \).

Then, \( b - a = -km \), which is also divisible by \( m \). Hence, \( b \equiv a \pmod{m} \).

Thus, the symmetry property holds.

### 3. Transitivity:
We need to show that if \( a \equiv b \pmod{m} \) and \( b \equiv c \pmod{m} \), then \( a \equiv c \pmod{m} \).

**Proof:**
Assume \( a \equiv b \pmod{m} \) and \( b \equiv c \pmod{m} \). This means that:
\[
a - b = km \quad \text{and} \quad b - c = lm
\]
for some integers \( k \) and \( l \).

Adding these two congruences, we get:
\[
(a - b) + (b - c) = km + lm
\]
\[
a - c = (k + l)m
\]

Since \( a - c \) is divisible by \( m \), it follows that \( a \equiv c \pmod{m} \).

Thus, the transitivity property holds.

### Conclusion:
Since congruence modulo \( m \) satisfies the properties of reflexivity, symmetry, and transitivity, it is indeed an equivalence relation.

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Do you have any questions or would you like further details? Here are some related topics you might find interesting:

1. What are equivalence classes under congruence modulo \( m \)?
2. How is modular arithmetic used in number theory?
3. Can you explain the Chinese Remainder Theorem?
4. How do congruences relate to solving Diophantine equations?
5. What is the significance of prime moduli in modular arithmetic?

**Tip:** When working with modular arithmetic, always remember that operations like addition, subtraction, and multiplication are well-defined under congruence.

Understand and prove that congruence modulo m forms an equivalence relation. Explore the definition, properties (reflexivity, symmetry, transitivity), and applications in number theory. Suitable for advanced high school and college students.

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