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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 -

    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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