query: If n is a positive integer, then $n^3 + 5n$ is divisible by $6$.
positive_passages.docid: 27798
positive_passages.text: Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in \textbf{N}$
negative_passages.docid: 82774
negative_passages.text: Prove by mathematical induction that $n^3 - n$ is divisible by $3$ for all natural number $n$
negative_passages.docid: 38904
negative_passages.text: Prove that $(a + b + c)^{13}$ is divisible by $abc$ if $b|a^3$, $c|b^3$ and $a|c^3$.
negative_passages.docid: 129238
negative_passages.text: Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.
negative_passages.docid: 50927
negative_passages.text: If $m,n\in N$ Prove that there is such a positive integer k, such that $(\sqrt{m}+\sqrt{m+1})^n=\sqrt{k}+\sqrt{k+1}$
negative_passages.docid: 116054
negative_passages.text: The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer
negative_passages.docid: 9302
negative_passages.text: Questions about Divisibility of $2^n$ by $3$
negative_passages.docid: 66578
negative_passages.text: Prove by induction that $2^{2n} – 1$ is divisible by $3$ whenever n is a positive integer.
negative_passages.docid: 120290
negative_passages.text: Probability that $7^m+7^n$ is divisible by $5$
negative_passages.docid: 17194
negative_passages.text: $6^{(n+2)} + 7^{(2n+1)}$ is divisible by $43$ for $n \ge 1$
negative_passages.docid: 112541
negative_passages.text: Prove that $n^2 + 2^n$ is composite if $n\not\equiv3\pmod{6}$.
negative_passages.docid: 29963
negative_passages.text: $6$ divides $a+b+c$ if and only if $6$ divides $a^{3}+b^{3}+c^{3}$
negative_passages.docid: 120227
negative_passages.text: Proving $6^n - 1$ is always divisible by $5$ by induction
negative_passages.docid: 10191
negative_passages.text: Binomial theorem: divisibility by $n^2$
negative_passages.docid: 125855
negative_passages.text: Prove: The product of any three consecutive integers is divisible by $6$.
negative_passages.docid: 11216
negative_passages.text: Induction: How to prove that $ab^n+cn+d$ is divisible by $m$.
negative_passages.docid: 98443
negative_passages.text: Find positive integers such that $2n^3 + 5|n^4 +n+1$