query: Prove that each natural number has a multiple solely consisting of digits $0,1$
positive_passages.docid: 27371
positive_passages.text: A natural number multiplied by some integer results in a number with only ones and zeros
negative_passages.docid: 124890
negative_passages.text: Divisibility within a block of natural numbers
negative_passages.docid: 121161
negative_passages.text: Let S be the set of all real numbers in the interval (0; 1) whose decimal expansions contain only 0's, 4's and 8's. Prove that S is uncountable.
negative_passages.docid: 156831
negative_passages.text: Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number
negative_passages.docid: 86535
negative_passages.text: Prove quotient of two $N(0,1)$ is $\text{Cauchy}(0,1)$
negative_passages.docid: 10744
negative_passages.text: Prove that a number is divisible by 3 iff the sum of its digits is divisible by 3
negative_passages.docid: 121469
negative_passages.text: Proof of observations on natural numbers being expressed as differences of squares.
negative_passages.docid: 35515
negative_passages.text: Find the number of $(m+n)$ digit binary sequences with $m$ $0$'s and $n$ $1$'s such that no two $1$'s are adjacent
negative_passages.docid: 100565
negative_passages.text: Property of sup of a set of numbers
negative_passages.docid: 48838
negative_passages.text: Proving the number of $n$ length binary strings with no consecutive $1's$ $b_n$ is equal to $b_{n-1} + b_{n-2}$
negative_passages.docid: 33786
negative_passages.text: Proof that a certain fraction is always an integer
negative_passages.docid: 141227
negative_passages.text: How to show the integers have same cardinality as the natural numbers?
negative_passages.docid: 156660
negative_passages.text: The set of all strictly increasing sequences of natural numbers
negative_passages.docid: 69875
negative_passages.text: Prove the commutativity property of addition of natural numbers by induction
negative_passages.docid: 156383
negative_passages.text: Proving a combinatorial identity without double counting
negative_passages.docid: 59788
negative_passages.text: Let T = {1000,1001...,9999}. How many numbers have at least one digit that is 0...
negative_passages.docid: 22869
negative_passages.text: Prove $[0, 1]$ and $(0, 1]$ are equinumerous by use of bijection