n⁹ - 6n⁷ + 9n⁵ - 4n³ is divisible by 8640.

means show:

n³(n-2)(n+2)(n-1)²(n+1)²=k(2².3³.5)
spread of factors are:
n-2 once
n-1 twice
n trice
n+1 twice
n+2 once

so for any integer, n, there will be a factor divisible by 5

also, for any integer, n, there will be at least two factors divisible by 2 ie if n is even there will be 5 factors divisible by 2, if n is odd, there will be 5 factors divisible by 2

also, for any integer, n, there will be at least 3 factor divisible by 3
ie if n is a factor of 3, you have n trice

if mod n/3 = 1, you have n-1 twice and n+2 once

if mod n/3 = 2, you have n-2 once and n+1 twice

Proven

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