Posted by Satyam on Sunday, June 23, 2013 at 2:19am.
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.

Maths  MathMate, Sunday, June 23, 2013 at 5:43am
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^517*n^3+16*n
=(n4)*(n1)*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
n4=0 (mod 4)
n1=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.