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.