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for this question would the negation of it use existanial quanifiers or would it keep the uiversla ones

The statement you're trying to negate is:

"For all integers \( a \), \( b \), and \( c \), if \( a \mid b \) and \( a \nmid (3b^3 - c) \), then \( 3b \nmid c \)."

This is a universally quantified statement, which means its negation will indeed use existential quantifiers.

### Original statement:
\[
\forall a, b, c \in \mathbb{Z}, (a \mid b \wedge a \nmid (3b^3 - c)) \implies 3b \nmid c
\]

### Negating the statement:
The negation of a universally quantified statement is an existentially quantified statement. So, the negation would be:

\[
\exists a, b, c \in \mathbb{Z}, (a \mid b \wedge a \nmid (3b^3 - c)) \wedge 3b \mid c
\]

This means there exists integers \( a \), \( b \), and \( c \) such that \( a \) divides \( b \), \( a \) does not divide \( 3b^3 - c \), but \( 3b \) divides \( c \).

### Summary of steps:
- The universal quantifier (\( \forall \)) becomes an existential quantifier (\( \exists \)).
- The implication (\( \implies \)) becomes a conjunction (\( \wedge \)), but you negate the second part of the implication.

Would you like further details or help with another part of the problem?

### 5 Related Questions:
1. What is the process of negating a universally quantified statement?
2. How can the contrapositive of this statement be useful in a proof?
3. What is the significance of dividing versus not dividing in number theory?
4. Can you provide an example of values of \( a \), \( b \), and \( c \) that satisfy or violate the conditions?
5. How do logical implications and their negations differ when handling inequalities?

### Tip:
When negating a mathematical statement, be careful to reverse all logical operators, including changing \( \forall \) (for all) to \( \exists \) (there exists), and negating implications.

Prove the following statement, citing all theorems used: For all integers a, b, and c, if a divides b and a does not divide (3b^3 - c), then 3b does not divide c.

Solve a proof problem in number theory: If a divides b and a does not divide (3b^3 - c), then 3b does not divide c. Learn about divisibility conditions and properties of number theory.

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