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The image shows a problem asking to "Find the whole number solutions of the congruence \( 2x + 3 \equiv 3 \mod 4 \)." The options listed are:

- A) \( \{0, 3, 7, 11, 15, \dots \} \)
- B) \( \{1, 5, 9, 13, 17, \dots \} \)
- C) \( \{2, 6, 10, 14, 18, \dots \} \)
- D) \( \{3, 8, 12, 16, 20, \dots \} \)

Let's solve this congruence step-by-step:

### Step 1: Solve the congruence
We are given the equation:
\[
2x + 3 \equiv 3 \mod 4
\]
Subtracting 3 from both sides:
\[
2x \equiv 0 \mod 4
\]
This simplifies to:
\[
x \equiv 0 \mod 2
\]
which means that \(x\) must be an even number.

### Step 2: Check the solutions
\(x\) can take values \(0, 2, 4, 6, \dots\), so we need to check which of the given options contains these numbers.

Looking at the options:
- A) \( \{0, 3, 7, 11, 15, \dots \} \) includes 0 but also odd numbers.
- B) \( \{1, 5, 9, 13, 17, \dots \} \) contains odd numbers only.
- C) \( \{2, 6, 10, 14, 18, \dots \} \) contains even numbers.
- D) \( \{3, 8, 12, 16, 20, \dots \} \) includes odd and even numbers but does not start with 0.

Thus, **option C** is the correct answer because it contains all even numbers.

Would you like more details or have any further questions?

### Related Questions:
1. How do you solve linear congruences in general?
2. What does the term "modulo" mean in modular arithmetic?
3. How can you determine whether a congruence has a unique solution?
4. What are some properties of even and odd numbers in modular arithmetic?
5. Can you apply the same method to solve quadratic congruences?

**Tip:** When solving congruences, always try simplifying the equation step by step, and pay attention to patterns in modular systems like mod 4 or mod 5.

Find the whole number solutions of the congruence (2x + 3) ≡ 3 mod 4.

This problem focuses on solving the linear congruence 2x + 3 ≡ 3 mod 4. By simplifying step-by-step, we find the solution x ≡ 0 mod 2, indicating that x must be an even number. Learn more about the method and concepts of modular arithmetic, including linear congruences and related theorems.

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