query: Is $(a+1)^n-a^n$ ever divisible by $n$?
positive_passages.docid: 32166
positive_passages.text: Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 89175
negative_passages.text: Proof of $(a+b)^{n+1}$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
negative_passages.docid: 146439
negative_passages.text: Is every sufficiently large positive integer of the form $ab + ac + bc + 1$?
negative_passages.docid: 134348
negative_passages.text: When is $(p - 1)! + 1$ a power of $p$?
negative_passages.docid: 55437
negative_passages.text: When does $2^{n}$ divide $3^{n}-1$
negative_passages.docid: 69136
negative_passages.text: factorization of a^n+1?
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: 3611
negative_passages.text: How to prove $n^5 - n$ is divisible by $30$ without reduction
negative_passages.docid: 104965
negative_passages.text: Find $n$ such that polynomial is divisible
negative_passages.docid: 1837
negative_passages.text: Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
negative_passages.docid: 40565
negative_passages.text: How can I prove that one of $n$, $n+2$, and $n+4$ must be divisible by three, for any $n\in\mathbb{N}$
negative_passages.docid: 122577
negative_passages.text: Prove that $(n^3 - 1)n^3(n^3 + 1)$ is divisible by $504$
negative_passages.docid: 100305
negative_passages.text: Prove $2^n-1$ is divisible by $(2^m-1)^2$ iff...
negative_passages.docid: 33866
negative_passages.text: Show that $n$ does not divide $2^n - 1$ where $n$ is an integer greater than $1$?