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.