To solve this, we will use Theorem 3.28 and Theorem 3.30 to demonstrate that if $ab \equiv 0 \pmod{3}$, then $a \equiv 0 \pmod{3}$ or $b \equiv 0 \pmod{3}$.

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Theorem 3.28 (Properties of Congruence):

If $a \equiv b \pmod{n}$ and $c \equiv d \pmod{n}$, then:

1. $a + c \equiv b + d \pmod{n}$,
2. $a \cdot c \equiv b \cdot d \pmod{n}$.

This means congruences respect addition and multiplication.

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Theorem 3.30 (Divisibility and Congruence):

If $n \mid ab$, then $n \mid a$ or $n \mid b$ (where $n$ is a prime number).

This theorem essentially asserts that the divisibility of the product implies the divisibility of at least one factor, provided $n$ is prime.

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Proof using Theorems 3.28 and 3.30:

Let $ab \equiv 0 \pmod{3}$. This means: $3 \mid ab.$

From Theorem 3.30, since $3$ is a prime number, $3 \mid a$ or $3 \mid b$. This implies: $a \equiv 0 \pmod{3} \quad \text{or} \quad b \equiv 0 \pmod{3}.$

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Why we avoid the direct definition:

The direct definition of congruence would require explicitly expressing $ab = 3k$ for some integer $k$, and then splitting $a$ or $b$ as multiples of 3. By using Theorem 3.30, we directly leverage the divisibility rules for prime numbers, which simplifies the proof significantly.

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Conclusion:

If $ab \equiv 0 \pmod{3}$, then $a \equiv 0 \pmod{3}$ or $b \equiv 0 \pmod{3}$, as demonstrated using Theorem 3.28 and Theorem 3.30.

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Do you have any questions or want more details? Here are 5 questions related to this topic:

1. Can you apply Theorem 3.30 to products modulo $n$ when $n$ is not prime?
2. How does Theorem 3.28 ensure congruence works under multiplication and addition?
3. What changes in the proof if the modulus is not a prime number, e.g., $ab \equiv 0 \pmod{6}$?
4. Can you derive Theorem 3.30 from the definition of congruence and divisibility?
5. How do Theorems 3.28 and 3.30 help simplify congruence proofs compared to direct definitions?

Tip: Always verify whether the modulus in your problem is prime. This simplifies divisibility reasoning significantly when applying modular arithmetic.