Show that if $n$ is not a multiple of $2$ then it is not a multiple of $6$

Question

I was trying to do this question and got a bit confused, am I to use quantifiers?

I came up with an answer that seems right but probably isn't.

Show that if $n$ is not a multiple of $2$ then it is not a multiple of $6$:

$$ \exists n \forall x [(n \neq 2x) \land (n \neq 6x)] $$

So I am trying to say that for some $n$, $n$ is not a multiple of $2$ and therefore is not a multiple of $6$.

Am I correct?

Any help is greatly appreciated, thank you in advance.

Answer 1

If $n$ is a multiple of $6$ then $n = 6k = 2(3k)$ is a multiple of $2$. This is proof by contrapositive

Answer 2

You want to prove the statement $\color\red{2\nmid n}\implies\color\green{6\nmid n}$.

Instead, prove the equivalent statement $\neg(\color\green{6\nmid n})\implies\neg(\color\red{2\nmid n})$.

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$\neg(6\nmid n)\implies$

$6\mid n\implies$

$\exists{k\in\mathbb{Z}}:[n=6\cdot k]\implies$

$\exists{k\in\mathbb{Z}}:[n=2\cdot(3\cdot k)]\implies$

$\exists{m\in\mathbb{Z}}:[n=2\cdot m]\implies$

$2\mid n\implies$

$\neg(2\nmid n)$