Question

Find the largest integer S which is a divisor of n^5−17n^3+16n for every integer n≥4. Tried it for a number of times. Please answer if you know how to solve it.

Answer 1

In the given case, since the given polynomial is of the fifth degree, you only have to try S=1,2,3,4 and up to 5.

In fact:
Let's factor the polynomial
f(x)=n^5-17*n^3+16*n
=(n-4)*(n-1)*n*(n+1)*(n+4)

From this you will find the factors have the following properties:
if
n=0 (mod 4)
then
n=0 (mod 4) ... hypothesis
n-4=0 (mod 4)
n-1=3 (mod 4)
n+1=1 (mod 4)
n+2=2 (mod 4) since n=0 (mod 4)

Which means that if S=4, then
f(n) = 0 for all n≥4 since the factors have residues 0,1,2,3 when divided by 4.