Use division algorithm and then induction to show 3|(n³+2n) for all ℕ.

Question

For division algorithm, would I do something along the lines of n³+2n = 3q+r and go from there?

For induction, I did the base case, which is true, and so then I moved on to the k+1 case, in which I did (n+1)³ + 2(n+1) to get n³+3n²+5n+3, which isn't all the way divisible by 3. WHat step did I do wrong here?

Answer 1

Assume that $3|n^3+2n$.

Then, we have $$(n+1)^3+2(n+1)=n^3+3n^2+5n+3=(n^3+2n)+3(n^2+n+1).$$ This is divisible by $3$ by the assumption.

Answer 2

Hint: $$ n^3+3n^2+5n+3 = (n^3 + 2n) + 3n^2 + 3n+ 3 $$