query: Is there a composite integer $n \geq 9$ such that $n \nmid (n-1)!$?
positive_passages.docid: 9319
positive_passages.text: If $n\ne 4$ is composite, then $n$ divides $(n-1)!$.
negative_passages.docid: 8034
negative_passages.text: Can $n!$ be a perfect square when $n$ is an integer greater than $1$?
negative_passages.docid: 41354
negative_passages.text: Binomial-coefficients if, k, m, n natural numbers and k \leq n the result of
negative_passages.docid: 97684
negative_passages.text: Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$
negative_passages.docid: 153072
negative_passages.text: Guessing a subset of $\{1,...,N\}$
negative_passages.docid: 142347
negative_passages.text: What is the highest power of $n$ in $(n^r-1)!$
negative_passages.docid: 139383
negative_passages.text: Prove that $\tau(2^n-1) \geq \tau(n)$ for all positive integers $n$.
negative_passages.docid: 2218
negative_passages.text: Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
negative_passages.docid: 143067
negative_passages.text: Show given any $x\in\mathbb{R}$ show there exists a unique $n\in\mathbb{Z}$ such that $n-1\leq x <n$.
negative_passages.docid: 124456
negative_passages.text: Given $x \in \mathbb{R}$ and $N \in \mathbb{N}, N>1,$ find integers $h,k$ with $0 < k \leq N $ that satisfy $\lvert{kx - h\rvert} < 1/N.$
negative_passages.docid: 2197
negative_passages.text: Induction principle for $n!<n^n$
negative_passages.docid: 34784
negative_passages.text: Is a number meeting these conditions divisible by forty-nine?
negative_passages.docid: 145891
negative_passages.text: Prove that $n! > \sqrt{n^n}, n \geq 3$
negative_passages.docid: 90806
negative_passages.text: Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
negative_passages.docid: 55266
negative_passages.text: Prove that $(n+1)^{n-1}<n^n$
negative_passages.docid: 149549
negative_passages.text: Infimum of $n$-th root of $n$
negative_passages.docid: 67208
negative_passages.text: Can we say that there exist an integer $n$ such $A+nB$ invertible?