To analyze this, let's break it down step by step. Recall that modular arithmetic has specific properties:

1. If $a \equiv b \pmod{m}$, this means $m \mid (a - b)$, or $a - b$ is divisible by $m$.
2. If $c \equiv d \pmod{m}$, this means $m \mid (c - d)$, or $c - d$ is divisible by $m$.

Using these properties, let's evaluate each option:

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A. $a + b \equiv c + d \pmod{m}$

• Adding congruences: $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$.
• This implies $a + c \equiv b + d \pmod{m}$, so this is true.

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B. $a - c \equiv b - d \pmod{m}$

• Subtracting congruences: $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$.
• This implies $a - c \equiv b - d \pmod{m}$, so this is true.

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C. $ac \equiv bd \pmod{m}$

• Multiplying congruences: $a \equiv b \pmod{m}$ and $c \equiv d \pmod{m}$.
• This implies $ac \equiv bd \pmod{m}$, so this is true.

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D. $ac \equiv bd \pmod{m}$

• This statement is identical to C, so this is also true.

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Final Answer:

All statements, A, B, C, and D, are true.

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Would you like a more detailed explanation for any specific part of this? Here are five related questions you might consider:

1. Can you provide examples to verify these modular properties?
2. How does modular arithmetic apply in real-world scenarios, like cryptography?
3. What are the limitations of modular arithmetic when performing operations like division?
4. Can you prove why the multiplication property holds in modular arithmetic?
5. How does the Chinese Remainder Theorem relate to these modular properties?

Tip: When solving modular problems, always check whether the modulus $m$ is consistent across all equations.