query: Factorial of a sum of n numbers divisible by product of factorial of these numbers
positive_passages.docid: 155150
positive_passages.text: Division of Factorials [binomal coefficients are integers]
negative_passages.docid: 8366
negative_passages.text: Proof of $a^n+b^n$ divisible by $a+b$ when $n$ is odd
negative_passages.docid: 43591
negative_passages.text: How to factor numbers that are the product of two primes
negative_passages.docid: 117761
negative_passages.text: When is $\binom{n}{k}$ divisible by $n$?
negative_passages.docid: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
negative_passages.docid: 20611
negative_passages.text: Inverse of a factorial
negative_passages.docid: 96086
negative_passages.text: Double Factorial: Number of possibilities to partition a set of $2n$ items into $n$ pairs
negative_passages.docid: 106888
negative_passages.text: Factorial identity $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+\cdots$
negative_passages.docid: 127855
negative_passages.text: Show that the product of any $m$ consecutive positive integers is divisible by $m!$
negative_passages.docid: 57986
negative_passages.text: Sum of products of combinatorials
negative_passages.docid: 85817
negative_passages.text: Show that in any set of $2n$ integers, there is a subset of $n$ integers whose sum is divisible by $n$.
negative_passages.docid: 90806
negative_passages.text: Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
negative_passages.docid: 78314
negative_passages.text: Generating function for the factorial sequence
negative_passages.docid: 17638
negative_passages.text: Sum with binomial coefficients: $\sum_{k=0}^{n}{2n\choose 2k}$
negative_passages.docid: 120913
negative_passages.text: How would you write a non-recursive algorithm to calculate factorials?
negative_passages.docid: 124890
negative_passages.text: Divisibility within a block of natural numbers
negative_passages.docid: 134921
negative_passages.text: Find the number of all subsets of $\{1, 2, \ldots,2015\}$ with $n$ elements such that the sum of the elements in the subset is divisible by 5