Largest number that divides $n^2(n^2 - 1)(n^2 - n - 2)$ for all $n$ Obtain the greatest natural that divides $n^2(n^2 - 1)(n^2 - n - 2)$ for all natural numbers $n$.
What should be the approach in these type of questions? 
Should I equate with prime factorization $2^a 3^b 5^c \cdots$ etc?

Factor as $(n-2)(n-1)n^2(n+1)^2$.  It is easy to see that no prime greater than $3$ must divide this product.  (For if $p$ is a prime greater than $3$, let $n-3=p$.)
Our product is obviously divisible by $3$. By choosing $n-1=3$, we can make sure that $3^2$ does not divide our product.
To minimize the power of $2$ that divides our product, we make $n-2$ (respectively, $n-1$) divisible by $4$ but not by $8$. Then $n^2$ (respectively, $(n+1)^2$) is divisible by $4$ but not by $8$. Thus $2^4$ must divide our product, and a higher power of $2$ need not.
We conclude that the largest positive integer that always divides our product is $(3^1)(2^4)$.
