query: For which $n\in\mathbb{N}$ is $(n+2)\mid(n^3+14)$ true?
positive_passages.docid: 40896
positive_passages.text: What is the largest positive $n$ for which $n^3+100$ is divisible by $n+10$
negative_passages.docid: 117681
negative_passages.text: For which Natural $n\ge2: \phi(n)=n/2$
negative_passages.docid: 143069
negative_passages.text: What are the direct summands in $\mathbb{Z}^n$?
negative_passages.docid: 63758
negative_passages.text: $c\mid ab\Rightarrow c\mid \gcd(a,c)\gcd(b,c),\,$ for integers $a, b, c$
negative_passages.docid: 84017
negative_passages.text: Is $\sqrt[3]{2}$ contained in $\mathbb{Q}(\zeta_n)$?
negative_passages.docid: 52951
negative_passages.text: Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
negative_passages.docid: 44897
negative_passages.text: Proof: If $n=ab$ then $2^a-1 \mid 2^n-1$
negative_passages.docid: 100558
negative_passages.text: Prove that for any even positive integer $n$, $n^2-1 \mid 2^{n!}-1$
negative_passages.docid: 26011
negative_passages.text: To show that $2$ | $x^{2n} - x^n$, $n \in \mathbb{Z^+}$ is the following adequate?
negative_passages.docid: 13682
negative_passages.text: Is $\langle a,b \mid a^2b^2=1 \rangle$ a semidirect product of $\mathbb{Z}^2$ and $\mathbb{Z}_2$?
negative_passages.docid: 33344
negative_passages.text: A different way to show that $a^k-1 \mid a^n-1$ if $k\mid n$?
negative_passages.docid: 97666
negative_passages.text: Prove $133\mid \left(11^{n+2}+12^{2n+1}\right)$
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: 30404
negative_passages.text: Is $\mathbb{Z}/3\otimes \mathbb{Z}[1/2]=\mathbb{Z}/3$?
negative_passages.docid: 48251
negative_passages.text: Prove that if $n \in \mathbb{N}$, then $\sum_{d|n}{(d(n))^3}=(\sum_{d|n}{d(n)})^2$ where $d(n)$ is the divisor function.
negative_passages.docid: 146093
negative_passages.text: Why if $n \mid m$, then $(a^n-1) \mid (a^m-1)$?
negative_passages.docid: 52565
negative_passages.text: Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.