This question is from Challenge and Thrill of Pre-College Mathematics.
Prove that $2n^3-3n^2+n$ is divisible by 6 for any natural number $n$.
I know there are many proofs of this using induction and many other ways. Is there any proof using concept of polynomials particularly using remainder theorem and factorization of polynomials? These concepts were introduced just before the exercise and the whole exercise was based on divisibility of a polynomial by another polynomial. Any hint/suggestions are really appreciated. Thank you!
I tried to prove that $2n^3-3n^2+n$ is divisible by $2b_1(n)$ and $3b_2(n)$ where $b_1(n)$ and $b_2(n)$ are polynomials over $\mathbb Z$ but could not found such polynomials.