query: Prove that $1^n+2^n+...+(n-1)^n$ is divisible by $n$ if $n$ is odd?
positive_passages.docid: 157294
positive_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: 11216
negative_passages.text: Induction: How to prove that $ab^n+cn+d$ is divisible by $m$.
negative_passages.docid: 140294
negative_passages.text: prove that $n >2$ and $n$ is a prime then $n$ is odd
negative_passages.docid: 51858
negative_passages.text: Showing $\gcd(2^m-1,2^n+1)=1$
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
negative_passages.docid: 4077
negative_passages.text: Using induction prove $n^3-n$ is divisible by 3 whenever n is a positive number.
negative_passages.docid: 85941
negative_passages.text: Proof by induction that $n^3 - n$ is divisible by $6$
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: 513
negative_passages.text: If n is an odd integer, show there exists a positive integer k such that 2^k mod n = 1.
negative_passages.docid: 31344
negative_passages.text: Proving $2^{2^n}+3^{2^n}+5^{2^n}$ is divisible by $19$ for all $n\geq 1$ by induction
negative_passages.docid: 33996
negative_passages.text: Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$
negative_passages.docid: 156641
negative_passages.text: Show that $\gcd(3n,3n+ 2) = 1$ when $n$ is odd
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: 36906
negative_passages.text: Prove that $x^n+x^{-n} \in \mathbf{N}$ if $x+\frac1x \in \mathbf{N}$
negative_passages.docid: 70625
negative_passages.text: Prove By Induction $1 + 1/2 + 1/3 + ... + 1/(2^n) ≥ 1 + n/2$
negative_passages.docid: 26336
negative_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 119436
negative_passages.text: Proving $2^n\leq 2^{n+1}-2^{n-1}-1$ for all $n\geq 1$ by induction