query: Proving that $8\mid n(n^{2}-1)(3n+2)$
positive_passages.docid: 16407
positive_passages.text: $24\mid n(n^{2}-1)(3n+2)$ for all $n$ natural problems in the statement.
negative_passages.docid: 72639
negative_passages.text: Show that $\sum_{k=0}^n\binom{3n}{3k}=\frac{8^n+2(-1)^n}{3}$
negative_passages.docid: 27922
negative_passages.text: prove that $\binom{n}{0}^2+\binom{n}{1}^2+\binom{n}{2}^2+\cdots+\binom{n}{n}^2=\binom{2n}{n}$
negative_passages.docid: 145041
negative_passages.text: Given that $p$ is a prime and $p\mid a^n$, prove that $p^n\mid a^n$.
negative_passages.docid: 138894
negative_passages.text: Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
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: 96152
negative_passages.text: Showing that $a^n - 1 \mid a^m - 1 \iff n \mid m$
negative_passages.docid: 157294
negative_passages.text: Prove that if $n>1$ is odd, then $1^n+2^n+3^n+\ldots+(n-1)^n$ is divisible by $n$
negative_passages.docid: 6165
negative_passages.text: Proof by Induction $n^2+n$ is even
negative_passages.docid: 144447
negative_passages.text: Show that $504 \mid (n^9 − n^3 )$ for any integer $n$
negative_passages.docid: 66703
negative_passages.text: Show that $8 \mid (a^2-b^2)$ for $a$ and $b$ both odd
negative_passages.docid: 90916
negative_passages.text: Prove that if $3\mid a^2+b^2$ then $3\mid a$ and $3\mid b$.
negative_passages.docid: 113119
negative_passages.text: Use the division algorithm to prove that 3|(n³ + 2n) for all n ∈ ℕ
negative_passages.docid: 53857
negative_passages.text: Showing $\sup \{ \sin n \mid n\in \mathbb N \} =1$
negative_passages.docid: 34906
negative_passages.text: Showing $[\mathbb{Q}(\sqrt[4]{2},\sqrt{3}):\mathbb{Q}]=8$.
negative_passages.docid: 66016
negative_passages.text: Prove $3^{2n+1} + 2^{n+2}$ is divisible by $7$ for all $n\ge0$
negative_passages.docid: 50614
negative_passages.text: If $4\mid a+bc$ and $6\mid b+ac$ prove that $2\mid a^2-b^2$