# 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.

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.

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