Re: [代數] 請教大大一題倍數證明
※ 引述《rfvbgtsport (uygh)》之銘言:
: 如何証明,連續n個正整數乘積,必為
: n!的倍數?
: 請大大指點一下,謝謝
令此 n 個連續正整數為 k+1,k+2,...,k+n, k 為不小於 0 之整數
並令 P = (k+1)(k+2)...(k+n)
則 k! * P = (k+n)!,又 P/n! = (k+n)!/(k!n!) 為整數
故 n!| P
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這時候我們不會乘以 3!,而是乘以 5!
=> 5! (6*7*8) = 8! 所以 6*7*8 = 56 * 3!
這樣說好了,讓這一堆連續正整數為 a_1,a_2,...,a_n
k := min {a_1,a_2,...,a_n} - 1
所以 k 是隨著你給定的整數不同就會跟著變的
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※ 編輯: Eliphalet (114.46.213.135), 05/29/2015 18:02:45
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就 C(n+k,k) 啊 @@
※ 編輯: Eliphalet (114.46.213.135), 05/29/2015 20:59:48
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我是覺得組合數是正整數是很自然的事情
那或者照你說的從 Pascal's triangle 下手
從 C(n+k,k) = C(n+k-1,k-1) + C(n+k-1,k)
用歸納法也可以證明是整數吧
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※ 編輯: Eliphalet (114.46.213.135), 05/29/2015 22:53:51
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