Let $p$ be a prime number. Prove that $p^{k}\choose n$ is divisible by $p$ for any $n$ where $1\le{n}\le{p^{k}}-1$ and $k$ is a positive integer.
121890
Proof that $ p $ | ${ p^n \choose k } $ for any prime $p$ and $ k < p^n$
142534
Proof of $k {n\choose k} = n {n-1 \choose k-1}$ using direct proof
64148
If $p$ and $q$ are distinct primes and $a$ be any integer then $a^{pq} -a^q -a^p +a$ is divisible by $pq$.
96062
Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
154305
Let $\pi$ denote a prime element in $\mathbb Z[i], \pi \notin \mathbb Z, i \mathbb Z$. Prove that $N(\pi)=2$ or $N(\pi)=p$, $p \equiv 1 \pmod 4$
91921
Divisibility of $n\cdot2^n+1$ by 3.
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.
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
70463
Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
135892
Understanding the proof for showing that there are infinitely many $n\in\mathbb{Z}^+$ such that $n!+1$ is divisible by at least two distinct primes
157884
For any prime $p>3$ show that 3 divides $2p^2+1$
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
24446
Prove that for each $n$, there exist $n$ consecutive integers, each of which is divisible by a perfect square larger than $1$.
158106
Prove that 1 less than the number of equivalence classes divides $p-1$ where $p$ is prime
71625
Divisibility question