Proving that if n is uneven, then n+5 is even

Question

I would think that of the five answers on the multiple choice the following one marked in bold is the correct one:

Answer 5.1: ∃k ∈ Z: n = 2(k + 1) ⇒ ∃k ∈ Z: n + 5 = 2(k + 1) + 5 = 2(k + 3) + 1

Answer 5.2: ∃k ∈ Z: n + 5 = 2k ⇒ ∃k ∈ Z: n = 2k − 5 = 2(k − 3) + 1

Answer 5.3: ∃k ∈ Z: n = 2k + 1 ⇒ ∃k ∈ Z: n + 5 = 2k + 6 = 2(k + 3)

Answer 5.4: ∃k ∈ Z: n = 2k ⇒ ∃k ∈ Z: n + 5 = 2k + 5 = 2(k + 2) + 1

Answer 5.5: ∃k ∈ Z: n + 5 = 2k + 1 ⇒ ∃k ∈ Z: n = 2k − 4 = 2(k − 2)

Answer 1

Translate them into plain language. $\exists k \in \mathbb Z : n = 2k$ means n is even and $\exists k \in \mathbb Z : n = 2k + 1$ means n is odd. The five choices are then

1. If n is even, then n+5 is odd
2. If n+5 is even, then n is odd
3. If n is odd, then n+5 is even
4. If n is even, then n+5 is odd
5. If n+5 is odd, then n is even

From this, it is clear that the third option is correct.