Question

Use division Algorithm to do the proof please show step by step with explain for each step

Answer 1

I'm not using algorithm. The proof doesn't need division algorithm.

Let's prove :-

If n(n²+2) is a multiple of 3 then,

n or n²+2 is a multiple of 3.

For when n is NOT a multiple of 3

Let n=3k+1

$n^2+2=9k^2+9k+3$

$n^2+2$ is a multiple of 3

Let n=3k+2

$n^2+2=9k^2+12k+3$

$n^2+2$ is a multiple of 3

Now, all possiblities are proven true.

Hence shown.