For all integers a, b, and c, if a | b and a ∤ (3b
3 − c), then 3b ∤ c.

Let's break down the logic behind this mathematical statement:

### Given:

1. \( a \mid b \): This means that \( a \) divides \( b \), i.e., \( b = ka \) for some integer \( k \).
2. \( a \nmid (3b^3 - c) \): This means that \( a \) does not divide \( 3b^3 - c \), i.e., \( 3b^3 - c \) is not divisible by \( a \).
3. We are asked to prove that \( 3b \nmid c \): This means that \( 3b \) does not divide \( c \), i.e., \( c \) is not a multiple of \( 3b \).

### Approach:

We are given that \( a \mid b \), meaning that \( b = ka \) for some integer \( k \). This fact will help in reasoning about the divisibility properties later.

From the second condition, \( a \nmid (3b^3 - c) \), we can rewrite it as:

\[
3b^3 - c \not\equiv 0 \pmod{a}
\]

In other words, \( 3b^3 \not\equiv c \pmod{a} \).

Now, if we assume that \( 3b \mid c \), then \( c = 3bm \) for some integer \( m \). Substituting this into the expression \( 3b^3 - c \), we get:

\[
3b^3 - c = 3b^3 - 3bm = 3b(b^2 - m)
\]

Thus, \( 3b^3 - c = 3b(b^2 - m) \).

If \( 3b \mid c \), this would imply that \( a \mid (3b^3 - c) \), because \( 3b^3 - c \) would be divisible by \( 3b \), and since \( a \mid b \), it would follow that \( a \mid (3b(b^2 - m)) \). However, this contradicts the given condition that \( a \nmid (3b^3 - c) \).

### Conclusion:

Therefore, our assumption that \( 3b \mid c \) must be false. Hence, we can conclude that:

\[
3b \nmid c
\]

Would you like a more detailed explanation or have any questions?

### Related Questions:
1. If \( a \mid b \), does \( a \mid 3b^3 \)?
2. Can \( c \) be a multiple of \( a \) given these conditions?
3. What happens if \( a = 1 \)?
4. How does the condition \( a \nmid (3b^3 - c) \) influence divisibility?
5. How does the relationship between \( a \) and \( b \) affect \( c \)?

### Tip:
When dealing with divisibility, always carefully check assumptions about factors and how modifying one term can affect the overall divisibility in the equation.

For all integers a, b, and c, if a | b and a ∤ (3b^3 − c), then 3b ∤ c.

Explore a detailed proof involving divisibility and modular arithmetic: if a | b and a ∤ (3b^3 - c), then 3b ∤ c. This problem demonstrates key number theory concepts and uses the method of contradiction to establish the conclusion. Suitable for advanced high school and undergraduate mathematics students.

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