Proofs using the Euclidean division algorithm

Question

How can I prove the following using the Euclidean division algorithm?

1. Every $n^4 + 4 + 11$ is divisible by 16, where n is odd. (n ∈ N)
2. $n(n+1)(n+2)$ is divisible by 24, where n is even. (n ∈ N)
3. $(n^2 - 1)$ is divisible by 6, where 6 and n are co-prime.

Answer 1

$(1)$ Using Carmichael function, $n^4\equiv1\pmod{16}$

$(2)$

First of all, product of $3$ is consecutive integer is always divisible by $3$

Let $n=2r$ where $r$ is an integer

$\displaystyle n(n+1)(n+2)=2r(2r+1)(2r+2)=8\cdot\frac{r(r+1)}2(2r+1)$ which is divisible by $8$

and lcm$(3,8)=24$

$(3)$If $(n,6)=1$ we can write $n=6m\pm1$ where $m$ is an integer as $(n,2)=1$

Answer 2

Since (1) and (3) have been answered, here's an idea for 2:

For even $n$, one of $n$ and $n + 2$ is divisible by 4 and the other by 2, so $n(n + 2)$ is divisible by 8. For any $n$, one of $n$, $(n + 1)$ and $(n + 2)$ is divisible by 3. Try to think of why and how to put these together.