query: Prove that for $n \geqslant 1,$ the integer $n(n+1)(2 n+1)$ is divisible by 6
positive_passages.docid: 74512
positive_passages.text: How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
negative_passages.docid: 17660
negative_passages.text: Show that the odd prime divisors of $n^2+n+1$ are of form $6k+1$ (exclude 3)
negative_passages.docid: 11168
negative_passages.text: Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
negative_passages.docid: 4315
negative_passages.text: show that if $n\geq1$, $(1+{1\over n})^n<(1+{1\over n+1})^{n+1}$
negative_passages.docid: 101927
negative_passages.text: Induction: Prove that $4^{n+1}+5^{2 n - 1}$ is divisible by 21 for all $n \geq 1$.
negative_passages.docid: 63416
negative_passages.text: Prove $(n^5-n)$ is divisible by 5 by induction.
negative_passages.docid: 129238
negative_passages.text: Divisibility of $2^n - 1$ by $2^{m+n} - 3^m$.
negative_passages.docid: 48333
negative_passages.text: Prove by induction that $(x+1)^n - nx - 1$ is divisible by $x^2$
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: 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: 145519
negative_passages.text: Prove that for any natural number $n>1$, we have $(2n)!<(n(n+1))^n$
negative_passages.docid: 27798
negative_passages.text: Prove that $ n^3 + 5n$ is divisible by 6 for all $n\in \textbf{N}$
negative_passages.docid: 68561
negative_passages.text: How to show $\lceil ( \sqrt3 +1)^{2n}\rceil$ where $n \in \mathbb{ N}$ is divisible by $2^{n+1}$
negative_passages.docid: 127833
negative_passages.text: Proving that $7^n(3n+1)-1$ is divisible by 9
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
negative_passages.docid: 102334
negative_passages.text: How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
negative_passages.docid: 108382
negative_passages.text: Prove $x^n-1$ is divisible by $x-1$ by induction