Write the contrapositive of "if $n$ is an even integer such that n + 1 is a square then n is divisible by 8"

Question

For the purpose of clarity, I do not want anybody to prove this statement, I am just looking to get some help translating it into the contrapositive using symbolic logic. Right now I have the following translation:

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Original Statement:

$$(\exists k \in \mathbb{Z} \quad n = 2k) \quad \land \quad (\exists m \in \mathbb{Z} \quad n+1 =m^2) \quad \Rightarrow \quad \text{n is divisible by 8}$$

Contrapositive Statement:

$$ \text{n is not divisible by 8} \quad \Rightarrow \quad (\exists k \in \mathbb{Z} \quad n = 2k+1) \quad \lor \quad (\forall m \in \mathbb{Z} \quad n+1 \neq m^2) $$

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A couple of points of clarification:

• I wasn't sure how to write $n$ is divisible by 8. Perhaps it could have been stated as $\exists p \in \mathbb{Z} \quad n = 8p$?
• For the negation of $\exists k \in \mathbb{Z} \quad n = 2k$, I know that typically you might want to reverse the $\exists$ to an $\forall$, but I think what I wrote makes sense since the negation of even numbers is odd numbers.

Eventually I will try to prove this statement by contrapositive, but I want to get some practice in with translating mathematical statements into a more formal structure.

My question(s) are following:

1. Is the original translation correct? How might it be improved?
2. How could I more formally translate '$n$ is divisible by 8'?
3. Is the contrapositive stated correctly?

I also am not able to use $mod$ notation, $a|b$ notation, or anything that is unique to Abstract Algebra.

Answer 1

If $n$ is not divisible by 8, then either $n$ is not an even integer or $n+1$ is not a square.

Answer 2

"if $n$ is an even integer such that $n+1$ is square then n is divisible by 8"

The original statement can be written as:

$$ (\forall k \in \mathbb{Z} \quad [n = 2k \quad \land \quad n \neq2k+1]) \quad \land \quad (\exists m\in \mathbb{Z}\quad m^2 = n+1) \quad \Rightarrow \quad 8 |n $$

The contrapositive is then:

$$ 8 \nmid n \quad \Rightarrow \quad (\exists k\in \mathbb{Z} \quad [n \neq 2k \quad \lor \quad n = 2k+1]) \quad \lor \quad (\forall m \in \mathbb{Z} \quad m^2 \neq n +1) $$