query: $\frac{(mn)!}{m!(n!)^m}\in\mathbb N^*$?
positive_passages.docid: 144836
positive_passages.text: How to prove that $\frac{(mn)!}{m!(n!)^m}$ is an integer?
negative_passages.docid: 146827
negative_passages.text: show that $\limsup(\frac{n!}{n^n})=0$
negative_passages.docid: 80216
negative_passages.text: Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
negative_passages.docid: 87843
negative_passages.text: Combinatorial proof that $\frac{({10!})!}{{10!}^{9!}}$ is an integer
negative_passages.docid: 96931
negative_passages.text: Supremum and infimum of $\{\frac{1}{n}-\frac{1}{m}:m, n \in \mathbb{N}\}$
negative_passages.docid: 83313
negative_passages.text: Showing that $\frac{\sqrt[n]{n!}}{n}$ $\rightarrow \frac{1}{e}$
negative_passages.docid: 137422
negative_passages.text: $\mathrm{card}(\mathbb{Z}^n/M\mathbb{Z}^n) = |\det(M)|$?
negative_passages.docid: 98946
negative_passages.text: Prove that $(mn)!$ is divisible by $(n!)\cdot(m!)^n$
negative_passages.docid: 78852
negative_passages.text: If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
negative_passages.docid: 51653
negative_passages.text: Prove that $m+\frac{4}{m^2}\geq3$ for every $m > 0$
negative_passages.docid: 106698
negative_passages.text: $\mathbb N$ when given the metric $d(m,n)=\frac{1}{m}-\frac{1}{n}$
negative_passages.docid: 6278
negative_passages.text: The value of $\sum_{1\leq l< m <n}^{} \frac{1}{5^l3^m2^n}$
negative_passages.docid: 14581
negative_passages.text: How do we prove $n^n \mid m^m \Rightarrow n \mid m$?
negative_passages.docid: 129801
negative_passages.text: Finding a closed form for $\sum^{n}_{k=1} \frac{k}{(k+1)!} $
negative_passages.docid: 20504
negative_passages.text: $\forall m,n \in \Bbb N$ : $\ 56786730\mid mn(m^{60}-n^{60})$
negative_passages.docid: 132732
negative_passages.text: Estimate of n factorial: $n^{\frac{n}{2}} \le n! \le \left(\frac{n+1}{2}\right)^{n}$
negative_passages.docid: 29609
negative_passages.text: Limits involving factorials $\lim_{N\to\infty} \frac{N!}{(N-k)!N^{k}}$