query: Given n integers, show that there is a subset whose sum is divisible by n.
positive_passages.docid: 22154
positive_passages.text: Show that given a set of positive n integers, there exists a non-empty subset whose sum is divisible by n
negative_passages.docid: 97947
negative_passages.text: How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$?
negative_passages.docid: 20713
negative_passages.text: The sum of an uncountable number of positive numbers
negative_passages.docid: 7785
negative_passages.text: Cardinality of sum-set
negative_passages.docid: 130615
negative_passages.text: divisibility of binomial coefficients by primes
negative_passages.docid: 114873
negative_passages.text: Show that if $n$ is not divisible by $2$ or by $3$, then $n^2-1$ is divisible by $24$.
negative_passages.docid: 124890
negative_passages.text: Divisibility within a block of natural numbers
negative_passages.docid: 150430
negative_passages.text: Let $S={1, 2, 3, 4, ....., 2070}$ find the number of subsets of $S$ whose sum of elements in $S$ is divisible by 9
negative_passages.docid: 157294
negative_passages.text: Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
negative_passages.docid: 25338
negative_passages.text: Sum of $n$ integers from set of size $2n-1$ divides $n$
negative_passages.docid: 88628
negative_passages.text: Proving the total number of subsets of S is equal to $2^n$
negative_passages.docid: 62100
negative_passages.text: Show that $n^3-n$ is divisible by $6$ using induction
negative_passages.docid: 51638
negative_passages.text: pigeonhole principle divisibility proof
negative_passages.docid: 68328
negative_passages.text: Infinitely many $n$ such that $2^n-n$ is divisible by a prime
negative_passages.docid: 124685
negative_passages.text: How to show $(mn)!$ is divisible by $(m!)^n$?
negative_passages.docid: 80365
negative_passages.text: Prove that $n$ is divisible by $6$
negative_passages.docid: 132605
negative_passages.text: Finding the number of subsets of a set such that an element divides the succeeding element.