query: $n > 1$, consecutive integers $a, a+ 1, . . . , a+ (n−1)$. Prove that the product thereom is always a multiple of $n!$
positive_passages.docid: 1507
positive_passages.text: The product of $n$ consecutive integers is divisible by $n$ factorial
negative_passages.docid: 79219
negative_passages.text: Prove that $a^{2^n} \equiv 1 \pmod{2^{n+2}},$ where $a$ is an odd integer.
negative_passages.docid: 97656
negative_passages.text: Finding $a_n$ if $a_0=a_1=1,a_{n+1}=n(a_n+a_{n-1})\ \ (n\ge 1).$
negative_passages.docid: 9920
negative_passages.text: $(n^2 - 1)!$ is a multiple of $(n!)^n$
negative_passages.docid: 139383
negative_passages.text: Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
negative_passages.docid: 118471
negative_passages.text: Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
negative_passages.docid: 69921
negative_passages.text: Prove that: $(1+a_1)(1+a_2)...(1+a_n) \le 1 + S_n + (S_n)^2/2! + ... + (S_n)^n/n!$
negative_passages.docid: 93657
negative_passages.text: Proving “The sum of $n$ consecutive cubes is equal to the square of the sum of the first $n$ numbers.”
negative_passages.docid: 48081
negative_passages.text: Prove that for all $n \in \mathbb{N}$: $2^{n-1}(a^n + b^n) \ge (a + b)^n$
negative_passages.docid: 93085
negative_passages.text: How to prove the identity $\prod_{n=1}^{\infty} (1-q^{2n-1}) (1+q^{n}) =1 $ for $|q|<1$?
negative_passages.docid: 95889
negative_passages.text: Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
negative_passages.docid: 102524
negative_passages.text: $ 1^k+2^k+3^k+...+(p-1)^k $ always a multiple of $p$?
negative_passages.docid: 14815
negative_passages.text: How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer
negative_passages.docid: 47960
negative_passages.text: Given any $10$ consecutive positive integers , does there exist one integer which is relatively prime to the product of the rest ?
negative_passages.docid: 89523
negative_passages.text: Infinite product $(1+z)\prod_{n=1}^{+\infty}(1+z^{2^{n}})$
negative_passages.docid: 56624
negative_passages.text: Prove that $0!+1! + 2! + 3! + ..... + n!$ $\neq$ $p^\text{r}$, where $n \geqslant 3$ and $n$, $p$ and $r$ are three integers
negative_passages.docid: 94575
negative_passages.text: Inductive proof that there are infinitely many pairs of consecutive powerful integers?