query: Prove that there exists 5 consecutive numbers, each of which is divisible by a cube of a positive integer greater than 1.
positive_passages.docid: 24446
positive_passages.text: Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$.
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: 127855
negative_passages.text: Show that the product of any $m$ consecutive positive integers is divisible by $m!$
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: 8364
negative_passages.text: To show that Fermat number $F_{5}$ is divisible by $641$.
negative_passages.docid: 47799
negative_passages.text: Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$
negative_passages.docid: 108746
negative_passages.text: Prove that there exist two integers such that i - j is divisible by n.
negative_passages.docid: 123117
negative_passages.text: Is it possible to find $n-1$ consecutive composite integers
negative_passages.docid: 23427
negative_passages.text: How to prove there are no more positive integers that are products of 2 and 3 consecutive numbers?
negative_passages.docid: 142956
negative_passages.text: Prove that for any positive integer n there is a sequence of n consecutive composite numbers.
negative_passages.docid: 120227
negative_passages.text: Proving $6^n - 1$ is always divisible by $5$ by induction
negative_passages.docid: 154494
negative_passages.text: Prove that there exists a row or a column of the chessboard which contains at least √n distinct numbers.
negative_passages.docid: 147470
negative_passages.text: Divisibility of Consecutive Powers of 10
negative_passages.docid: 51638
negative_passages.text: pigeonhole principle divisibility proof
negative_passages.docid: 78593
negative_passages.text: Prove that the product of four consecutive positive integers plus one is a perfect square
negative_passages.docid: 101927
negative_passages.text: Induction: Prove that $4^{n+1}+5^{2 n - 1}$ is divisible by 21 for all $n \geq 1$.
negative_passages.docid: 49096
negative_passages.text: How can I prove $5^{3^n}+1$ is divisible by $3^{n+1}$