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This image presents a mathematical proposition and its corresponding demonstration involving congruences. Let me break it down step by step for clarity:

### Proposition:
Let \( m \in \mathbb{N} \) and \( a, b, c, d \in \mathbb{Z} \) such that:
\[
a \equiv b \pmod{m}, \quad c \equiv d \pmod{m}
\]
Then the proposition states that:
\[
ac \equiv bd \pmod{m}.
\]

### Demonstration:

1. **Given**: 
   \[
   a \equiv b \pmod{m}, \quad c \equiv d \pmod{m}
   \]
   By definition, this means there exist integers \( k_1 \) and \( k_2 \) such that:
   \[
   a = mk_1 + b, \quad c = mk_2 + d.
   \]

2. **Substitute the definitions** of \( a \) and \( c \) into the product \( ac \):
   \[
   ac = (mk_1 + b)(mk_2 + d).
   \]

3. **Expanding** this expression gives:
   \[
   ac = mk_1mk_2 + mk_1d + bmk_2 + bd.
   \]

4. Since \( mk_1mk_2 \), \( mk_1d \), and \( bmk_2 \) are all multiples of \( m \), it follows that:
   \[
   ac \equiv bd \pmod{m}.
   \]

Thus, the proposition is proved, showing that if \( a \equiv b \pmod{m} \) and \( c \equiv d \pmod{m} \), then \( ac \equiv bd \pmod{m} \).

---

Do you want more details or have any questions about this proof?

Here are 5 relative questions expanding on this topic:
1. What is the meaning of modular arithmetic in number theory?
2. How does the property of congruence extend to addition and subtraction?
3. Can this proposition be generalized to higher powers of \( a \) and \( c \)?
4. How would this result change if \( m \) were not a natural number but a prime?
5. Can you derive similar properties for division under modular arithmetic?

**Tip**: When dealing with modular arithmetic, always remember that the modulo operation respects addition, subtraction, and multiplication, making it useful for simplifying large numbers.

Sejam m ∈ N e a, b, c, d ∈ Z tais que a ≡ b mod m e c ≡ d mod m, então ac ≡ bd mod m.

Learn how to prove that if a ≡ b mod m and c ≡ d mod m, then ac ≡ bd mod m. This problem involves key concepts in modular arithmetic and congruences, suitable for advanced students studying number theory or abstract algebra.

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