Every integer of the form $(n^{3}-n)(n^{2}-4)$ (for n = 3,4,....K) is divisible by?

Question

Every integer of the form $(n^{3}-n)(n^{2}-4)$ $(for n = 3,4,....K)$ is

(A) divisible by $6$ but not always divisible by $12$;

(B) divisible by 12 but not always divisible by 24;

(C) divisible by 24 but not always divisible by 120;

(D) divisible by 120 but not always divisible by 720.

Simplifying the equation did not help, and after solving it for $3$ and $4$ we see that it is divisible by everything except $720$.

But how do we find out if this holds true for the entire sequence?

Answer 1

Hint $$(n^3-n)(n^2-4)=(n-2)(n-1)n(n+1)(n+2).$$ This is a product of five consecutive integers, hence divisible by $5!$