證明連續N個正整數的乘積能被N!整除
車五進二
Let p be any prime number between 1 and n.In n!, there are [n/p] consecutive multiples p, where [x] is the greatest integer less than or equal to x.
Therefore the number of factors of p in n! is [n/p] + [n/(p^2)] + [n/(p^3)]...
For N consecutive numbers, by the pigeon hole theorem, there are at least [n/p] consecutive multiples of p, therefore the number of factors of p in the product is at least [n/p] + [n/(p^2)] + [n/(p^3)]..., hence divisible by the factors of p in n!.
Since this applies to all the prime factors of n!, therefore K(K+1)...(K+N-1) is divisible by n!.
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