证明多项式整除n为任意整数,n^6-3n^5+6n^4-7n^3+5n^2-2n都能被24整除

Question

证明多项式整除
n 为任意整数,n^6-3n^5+6n^4-7n^3+5n^2-2n都能被24整除

Answer 1

x(x-1)(x-2)(x-3)≡0(mod4!)
x^4≡6x^3-11x^2+6x(mod24)
n^6-3n^5+6n^4-7n^3+5n^2-2n
≡3n^5-5n^4-n^3+5n^2-2n
≡13n^4-34n^3+23n^2-2n
≡-4n^3+4n≡-4n(n+1)(n-1)
≡-24C(n+1,3)≡0(mod24)
用待定系数法可得出x^n≡ax^3+bx^2+cx(mod24)