query: $(n-1)^2$ divides $(n^k -1) $ if and only if $(n-1)$ divides $k$
positive_passages.docid: 9997
positive_passages.text: For what powers $k$ is the polynomial $n^k-1$ divisible by $(n-1)^2$?
negative_passages.docid: 959
negative_passages.text: Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.
negative_passages.docid: 50534
negative_passages.text: Prove that if $n^2$ is divided by 3, then also $n$ can also be divided by 3.
negative_passages.docid: 136077
negative_passages.text: Prove that $x-1$ is a factor of $x^n-1$
negative_passages.docid: 31555
negative_passages.text: Divisor of $m$ and $n$ divides $m - qn$ (in proof of Euclidean algorithm)
negative_passages.docid: 108382
negative_passages.text: Prove $x^n-1$ is divisible by $x-1$ by induction
negative_passages.docid: 40131
negative_passages.text: If n is such that every element $(\mathbb{Z}/n\mathbb{Z})^{\times}$ is a root of $x^2-1$. Prove that $n$ divides 24.
negative_passages.docid: 112147
negative_passages.text: Proving $9$ divides $n^3 + (n+1)^3 + (n+2)^3$
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: 59479
negative_passages.text: For a positive integer $n\geq 2$ with divisors $1=d_1<d_2<\cdots<d_k=n$, prove that $d_1d_2+d_2d_3+\cdots+d_{k-1}d_k<n^2$
negative_passages.docid: 51706
negative_passages.text: If $(k,l) = 1$, show that $(b_2k-b_1l, a_1l-a_2k) = 1$ for $a_1b_2-a_2b_1=1$
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: 121723
negative_passages.text: $\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k
negative_passages.docid: 31933
negative_passages.text: $x^2 = 1$ in the ring $\mathbb{Z}/n \mathbb{Z}$
negative_passages.docid: 77076
negative_passages.text: Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$
negative_passages.docid: 124131
negative_passages.text: Show that $(x-1)^2$ is a factor of $x^n -nx +n-1$
negative_passages.docid: 120976
negative_passages.text: Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $