How can we prove that $\displaystyle \frac{n^2!}{(n!)^2}\in \mathbb{Z}\;,$ Where $n\in \mathbb{N}$
45645
Using permutation or otherwise, prove that $\frac{(n^2)!}{(n!)^n}$ is an integer,where $n$ is a positive integer.
5257
Proving that $(3n)!$ is divisible by $n! \times (n + 1)! \times (n + 2)!$ if $n$ is greater than 2
156766
Prove that $\sqrt{\frac{n}{n+1}}\notin \mathbb{Q}$
141837
Proof of $\sup\{ (n!)^{(\frac{1}{n})}| n ∈ N\} = +∞?$
152698
Evaluation of $\sum_{n=0}^{\infty}\frac{1}{(n^4+n^2+1)n!}$
137640
Proof that $\sum_{1}^{\infty} \frac{1}{n^2} <2$
52427
Prove, $\sqrt{n} \le (n!)^\frac{1}{n} \le \frac{n+1}{2} $
19804
How to prove that $|w_{n+1}-w_n| \leq (2/3)^2 |w_n - w_{n-1}|$?
68400
Proof that $\frac{(2n)!}{2^n}$ is integer
133451
To show that ${x_n}$ is convergent where $|x_{n+1} - x_n|< \frac{1}{n^2}$
132431
Proving $|f'(z)|\leq\frac{\text{Re}(f(z))}{\text{Re}(z)}$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
24469
Prove $ 2<(1+\frac{1}{n})^{n}$
20691
prove that $\frac{(2n)!}{(n!)^2}$ is even if $n$ is a positive integer
17241
How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$
9920
$(n^2 - 1)!$ is a multiple of $(n!)^n$
44928
Show $\sum_n \frac{z^{2^n}}{1-z^{2^{n+1}}} = \frac{z}{1-z}$