Prove some divisibility with deductive way

Question

I have some proves But I want to prove them with deductive not inductive.
Here are my proves:

1) $2^{3n} - 1 $ is divisible by 7.

2) $2^n + (-1)^{n+1}$ is divisible by 3.

3) $n^2 + 2$ is not divisible by 4.

4) $11^n - 4^n$ is divisible by 7.

Is it possible to help me? (some of them is good too).
Thanks.

Answer 1

1. $2^{3n} - 1 = 8^n-1 = (8-1)a = 7a$ where a is the other factor.
   Hence, $2^{3n} - 1 $ is divisible by 7.
   
2. $2^n + (-1)^{n+1} = 2^n - (-1)^n = [2-(-1)]b = 3b$ where b is the other factor.
   Hence,$2^n + (-1)^{n+1}$ is divisible by 3.
3. $n^2 + 2 = 4k + 2 = 2(2k+1) = 2 \times \text{odd number}$, not divisible by $4$ ,when $n$ is even
   $n^2 + 2 = 8k+1+2 = 8k+3$, not divisible by $4$, when $n$ is odd
   Hence, $n^2 + 2$ is not divisible by 4.
4. $11^n - 4^n = (11-4)c = 7c$ where c is the other factor.
   Hence, $11^n - 4^n$ is divisible by 7.