Prove $\forall x\in\mathbb{Z}, k!\mid x(x+1)(x+2)\cdots(x+k-1)$.
1507
The product of $n$ consecutive integers is divisible by $n$ factorial
145048
Prove $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^n = n!$ and $\sum_{k = 0}^{n}(-1)^{n - k} \binom{n}{k} \cdot k^m = 0$
74561
Let $a_1,a_2,\cdots,a_n$ be $n$ numbers such that $a_i$ is either $1$ or $-1$.If $a_1a_2a_3a_4+\cdots+a_na_1a_2a_3=0$ then prove that $4\mid n$.
17677
Prove or show a counter example for: $\forall a, b, c \in \mathbb Z$, if $ab\mid c$ then $a\mid c$ and $b\mid c$
73277
Prove $[\sqrt{x}+\sqrt{x+1}]=[\sqrt{4x+2}], x \in \Bbb N$
80216
Prove if $n \in \mathbb N$, then $\frac{1}{2!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$
52565
Prove that if $d \mid n \in \mathbb{N}$, then $\varphi(d) \mid \varphi(n)$.
33344
A different way to show that $a^k-1 \mid a^n-1$ if $k\mid n$?
138894
Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
34154
Prove the statement $\forall x\in\mathbb N (x > 1\to\exists k\in\mathbb N\exists m \in\mathbb N (m \equiv 1 (\text{mod }2) \wedge x = 2^km))$.
148458
$x^p-x \equiv x(x-1)(x-2)\cdots (x-(p-1))\,\pmod{\!p}$
141088
Proof : $2^{n-1}\mid n!$ if and only if $n$ is a power of $2$.
153763
Prove that $a_1!\cdots a_k! < n!$ whenever $a_1+\cdots+a_k < n$
10976
Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$
136459
Prove ${n - 1 \choose k - 1} + {n - 2 \choose k - 1} + {n - 3 \choose k - 1} + \dots + {k - 1 \choose k - 1} = {n \choose k}$
63553
Show that $\frac{1}{(n+1)!}(1+\frac{1}{(n+1)}+\frac{1}{(n+1)^2}+\frac{1}{(n+1)^3}+\cdots)=\frac{1}{n!n}$
115459
Proving this $F_{n+1} \cdot F_{n-1} - F^2_n = (-1)^n$ by induction