query: Prove that $2730$ divides $n^{13} - n$ for all integers $n$.
positive_passages.docid: 1895
positive_passages.text: How to show that $2730\mid n^{13}-n\;\;\forall n\in\mathbb{N}$
negative_passages.docid: 65430
negative_passages.text: Using the Well Ordering Principle to Show that n^3 - n is divisible by 6.
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: 3632
negative_passages.text: Basic number theory proofs
negative_passages.docid: 91921
negative_passages.text: Divisibility of $n\cdot2^n+1$ by 3.
negative_passages.docid: 113119
negative_passages.text: Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
negative_passages.docid: 80144
negative_passages.text: Prove that $x-1$ divides $x^n-1$
negative_passages.docid: 98946
negative_passages.text: Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$
negative_passages.docid: 64654
negative_passages.text: Suppose $a, b$ and $n$ are positive integers. Prove that $a^n$ divides $b^n$ if and only if $a$ divides $b$.
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 77816
negative_passages.text: Proof problem: Show that $n^2-1$ is divisible by $8$, if $n$ is an odd positive integer.
negative_passages.docid: 32166
negative_passages.text: Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$
negative_passages.docid: 59135
negative_passages.text: What is the proof of divisibility by $13$?
negative_passages.docid: 153313
negative_passages.text: Prove that for all integers $ a $ and $ b $, if $ a $ divides $ b $, then $ a^{2} $ divides $ b^{2} $.
negative_passages.docid: 115866
negative_passages.text: Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
negative_passages.docid: 72511
negative_passages.text: If $n$ is an odd natural number, then $8$ divides $n^{2}-1$
negative_passages.docid: 11867
negative_passages.text: Prove by induction that $n < 2^n$ for all $n \ge 1$