query: Give a proof by cases that shows that $n(n ^2 − 1)(n + 2) $ is a multiple of $4$, for all integers $n$
positive_passages.docid: 26336
positive_passages.text: Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
negative_passages.docid: 10917
negative_passages.text: Give a combinatorial proof: $ n(n+1)2^{n-2} = \sum_{k=1}^{n}k^2\binom{n}{k} $
negative_passages.docid: 74512
negative_passages.text: How to show $n(n+1)(2n+1) \equiv 0 \pmod 6$?
negative_passages.docid: 120976
negative_passages.text: Proving $\sum_{k=0}^{n}k{n\choose k}^2 = n{2n-1 \choose n-1} $
negative_passages.docid: 94923
negative_passages.text: Using the Pigeonhole Principle to show that $2$ of any $n+1$ numbers from $\{1,2,\ldots,2n\}$ sum to $2n+1$
negative_passages.docid: 100361
negative_passages.text: Show that ord$_{p}2 = 2^{n + 1}$.
negative_passages.docid: 25826
negative_passages.text: Proof of a combinatorial identity: $\sum\limits_{i=0}^n {2i \choose i}{2(n-i)\choose n-i} = 4^n$
negative_passages.docid: 21612
negative_passages.text: How to prove this form of $n$?
negative_passages.docid: 9435
negative_passages.text: Prove that $1^2+2^2+\cdots+n^2=\frac{n(n+1)(2n+1)}{6}$ for $n \in \mathbb{N}$.
negative_passages.docid: 73277
negative_passages.text: Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 77806
negative_passages.text: Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
negative_passages.docid: 55742
negative_passages.text: Showing that $f_{2n+1}=f_{n+1}^2+f_n^2$.
negative_passages.docid: 71534
negative_passages.text: Combinatorial explanation for why $n^2 = {n \choose 2} + {n+1 \choose 2}$
negative_passages.docid: 29828
negative_passages.text: Combinatorial proof of $\sum^{n}_{i=1}\binom{n}{i}i=n2^{n-1}$.
negative_passages.docid: 121544
negative_passages.text: Proof by Induction: $2(\sqrt{n+1} - \sqrt{n}) < \frac{1}{\sqrt{n}} < 2(\sqrt{n}-\sqrt{n-1})$
negative_passages.docid: 3632
negative_passages.text: Basic number theory proofs