Let $p$ be a prime. Show that ${n \choose p}-\bigl[\frac{n}{p}\bigr]$ is divisible by $p$, for all $n\in \mathbb{N}$
30630
Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
60804
About the property of $m$: if $n < m$ is co-prime to $m$, then $n$ is prime
90806
Find all positive integers $n$ such that $n$ is divisible by all the positive integers less than or equal to $\sqrt{n}$
95111
Elementary proof for $\sqrt{p_{n+1}} \notin \mathbb{Q}(\sqrt{p_1}, \sqrt{p_2}, \ldots, \sqrt{p_n})$ where $p_i$ are different prime numbers.
78356
How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$
138645
Prove Divisibility In Fibonacci Sequence Over A Prime Number
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
11470
Divisibility of integers
27170
$a\in \mathbb{N}$, $p$ prime, $a<p$ prove that $a\mid p+1\iff\exists\, b,c\in\Bbb N:\dfrac{a}{p}=\dfrac{1}{b}+\dfrac{1}{c}$
154076
How to show this property of $\mathbb{RP}^{n}$.
115004
If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?
154305
Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
27449
Product of Divisors of some $n$ proof
91921
Divisibility of $n\cdot2^n+1$ by 3.
33011
Prove that the polynomial $f_n(x)=nx^{n+1}-(n+1)x^n+1$ is divisible by $(x-1)^2$
153892
Number Theory divisibilty
77240
Show that $4 + \sqrt{10}$ is not prime in $\mathbb{Z}\bigl [\sqrt{10}\bigr]$