query: Let $S=\{1,2,3,4,5....,2070\}$ and $A$ be the number of subsets of $S$ whose sum of elements in $S$ is divisible by $9$
positive_passages.docid: 150430
positive_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: 117380
negative_passages.text: Divisibility Rule for 9
negative_passages.docid: 104527
negative_passages.text: Prove that there are two subsets of any cardinality-10 subsets of $\{1,2\dots,106\}$ that sum to the same number
negative_passages.docid: 60309
negative_passages.text: Let X be a subset of {1,2,...,2014}...
negative_passages.docid: 139078
negative_passages.text: Infinite partition of $\mathbb N$ by infinite subsets
negative_passages.docid: 25338
negative_passages.text: Sum of $n$ integers from set of size $2n-1$ divides $n$
negative_passages.docid: 104855
negative_passages.text: Prove that $N \subset Z(G)$
negative_passages.docid: 77200
negative_passages.text: Prove:that in any set of 1009 positive integers exits two numbers $a_i$ and $a_j$ such that $a_i-a_j$ or $a_i+a_j$ is divisible by 2014
negative_passages.docid: 146098
negative_passages.text: Number of nonempty subsets of a set with $n$ elements
negative_passages.docid: 74523
negative_passages.text: Proof by Cases involving divisibility
negative_passages.docid: 127833
negative_passages.text: Proving that $7^n(3n+1)-1$ is divisible by 9
negative_passages.docid: 147484
negative_passages.text: For a set with N members, what is the number of set partitions in which each subset is either of size 1 or 2?
negative_passages.docid: 107725
negative_passages.text: Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
negative_passages.docid: 51617
negative_passages.text: What is the proof that the total number of subsets of a set is $2^n$?
negative_passages.docid: 104010
negative_passages.text: The number of partitions of $n$ into distinct parts equals the number of partitions of $n$ into odd parts
negative_passages.docid: 114780
negative_passages.text: $\forall a,b,c \in \mathbb{N}\!:\ a\mid b \Rightarrow ac\mid bc. $ Divisibility Proof
negative_passages.docid: 136306
negative_passages.text: Let $A$ be a finite set and let $B$ be a subset of $A$ with $|A|=n$,$|B|=m$ and $0<m<n$. Count the subsets of $A$ containing $ B.$