query: Show that $504 \mid n^9 − n^3 $ for any integer $n$
positive_passages.docid: 144447
positive_passages.text: Show that $504 \mid (n^9 − n^3 )$ for any integer $n$
negative_passages.docid: 48053
negative_passages.text: Show that 13 divides $2^{70}+3^{70}$
negative_passages.docid: 10236
negative_passages.text: Show that $a^n \mid b^n$ implies $a \mid b$
negative_passages.docid: 26461
negative_passages.text: Integer solutions of $x^3-x+9=5y^2$
negative_passages.docid: 114867
negative_passages.text: Prove $\forall n\geq 2,n\in\mathbb{Z}$, $(n+1)\mid(n^3+1)$
negative_passages.docid: 95000
negative_passages.text: Using induction to prove that $2 \mid (n^2 − n)$ for $n\geq 1$
negative_passages.docid: 53162
negative_passages.text: $13\mid4^{2n+1}+3^{n+2}$
negative_passages.docid: 152870
negative_passages.text: Near integers in powers of binomials with radicals
negative_passages.docid: 46556
negative_passages.text: How to prove $3^{\log_4n} = n^{\log_43}$?
negative_passages.docid: 127293
negative_passages.text: Prove that $(a-b) \mid (a^n-b^n)$
negative_passages.docid: 86479
negative_passages.text: How to Prove with Mathematical Induction $3^n > n^2$
negative_passages.docid: 77383
negative_passages.text: Prove that $n^3=n \text{ mod }6$ for every integer $n$.
negative_passages.docid: 143543
negative_passages.text: Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
negative_passages.docid: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 113119
negative_passages.text: Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
negative_passages.docid: 19701
negative_passages.text: Prove $5\mid2^{4n}-1$ by induction
negative_passages.docid: 21100
negative_passages.text: Simple Proof by induction: $9$ divides $n^3 + (n+1)^3 + (n+2)^3$