query: Prove that $7\mid n^7+7n^3-n$ for every positive integer n.
positive_passages.docid: 24053
positive_passages.text: How can I prove that $n^7 - n$ is divisible by $42$ for any integer $n$?
negative_passages.docid: 39175
negative_passages.text: Prove by induction that $\forall n \in\mathbb N$, $3\mid4^{n}-1$.
negative_passages.docid: 81136
negative_passages.text: Prove by induction that... $1+3+5+7+...+(2n+1)=(n+1)^2$ for every $n \in \mathbb N$
negative_passages.docid: 14581
negative_passages.text: How do we prove $n^n \mid m^m \Rightarrow n \mid m$?
negative_passages.docid: 119118
negative_passages.text: Prove $n\mid \phi(2^n-1)$
negative_passages.docid: 53857
negative_passages.text: Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
negative_passages.docid: 138846
negative_passages.text: Prove by induction that $3\mid n^4-n^2$ for all $n \in \mathbb{Z^+}$, $n\ge2$.
negative_passages.docid: 11693
negative_passages.text: If $p \mid m^p+n^p$ prove $p^2 \mid m^p+n^p$
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: 139230
negative_passages.text: Prove that for every positive integer, this polynomial is divisible by 24.
negative_passages.docid: 27850
negative_passages.text: Find and prove a formula for $1\pmod{3}\cap 1\pmod{7}$.
negative_passages.docid: 87522
negative_passages.text: If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$
negative_passages.docid: 156786
negative_passages.text: Show that $15\mid 21n^5 + 10n^3 + 14n$ for all integers $n$.
negative_passages.docid: 16660
negative_passages.text: Prove that any positive number has an $n$th root
negative_passages.docid: 133819
negative_passages.text: How to prove that $364 \mid n^{91} - n^7$?
negative_passages.docid: 27798
negative_passages.text: Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in \textbf{N}$
negative_passages.docid: 78416
negative_passages.text: Show that $7$ divides $a^{15}-a^3$ for any integer $a$