N^3+6N^2+11N+6 一定是6的倍數(shù)嗎,說明理由

數(shù)學(xué)人氣：455 ℃時(shí)間：2020-09-24 22:36:04

優(yōu)質(zhì)解答

N^3+6N^2+11N+6
=N^3-N+6N^2+12N+6
=N(N+1)(N-1)+6(N+1)^2
=(N+1)(N^2-N+6N+6)
=(N+1)(N^2+5N+6)
=(N+1)(N+2)(N+3)
三個(gè)連續(xù)整數(shù)中至少有一個(gè)是偶數(shù)
所以(N+1)(N+2)(N+3)是2的倍數(shù)
且三個(gè)連續(xù)整數(shù)中一定有一個(gè)是3的倍數(shù)
所以(N+1)(N+2)(N+3)是3的倍數(shù)
2和3互質(zhì)
2*3=6
所以(N+1)(N+2)(N+3)是6的倍數(shù)