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The problem presented in the image involves proving a modular arithmetic property.

### Problem Statement:
Let \( n \) be a natural number, and let \( a, b, c, \) and \( d \) be integers. We are given two modular congruences:
- \( a \equiv b \pmod{n} \)
- \( c \equiv d \pmod{n} \)

We are asked to prove that:
\[
(a + c) \equiv (b + d) \pmod{n}
\]

### Proof:

#### Step 1: Understanding the modular equivalences

From \( a \equiv b \pmod{n} \), we know that \( a - b \) is divisible by \( n \), which means:
\[
a = b + kn \quad \text{for some integer} \, k
\]
Similarly, from \( c \equiv d \pmod{n} \), we know that \( c - d \) is divisible by \( n \), which means:
\[
c = d + ln \quad \text{for some integer} \, l
\]

#### Step 2: Add the two equations

Now, adding these two congruences:
\[
a + c = (b + kn) + (d + ln)
\]
Simplifying this:
\[
a + c = (b + d) + (k + l)n
\]

#### Step 3: Conclusion

From the equation \( a + c = (b + d) + (k + l)n \), we see that \( (a + c) - (b + d) = (k + l)n \), which means that \( a + c \equiv b + d \pmod{n} \).

Thus, we have proven the required result:
\[
(a + c) \equiv (b + d) \pmod{n}
\]

---

Do you have any questions or would you like more details on any of the steps? Here are a few related questions for further exploration:

1. How does modular arithmetic apply to cryptography?
2. Can you prove the result for multiplication: \( ac \equiv bd \pmod{n} \)?
3. What happens if \( n \) is not a natural number but an integer?
4. What are the real-world applications of modular arithmetic?
5. Can we extend this property to more than two numbers (i.e., summing multiple congruences)?

**Tip:** When working with modular arithmetic, always focus on the idea that congruent numbers differ by a multiple of the modulus.

Let n be a natural number, and let a, b, c, and d be integers. Prove that if a ≡ b (mod n) and c ≡ d (mod n), then (a + c) ≡ (b + d) (mod n).

Explore the proof for the additivity property in modular arithmetic: (a + c) ≡ (b + d) (mod n). This problem introduces key concepts in number theory, using modular equivalences to demonstrate congruence preservation under addition. Suitable for undergraduate students in discrete mathematics.

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