Given $p$ prime and $0<k<p$, show that $p$ divides ${p \choose k}$
63893
Prime dividing the binomial coefficients
144622
Proof: If no prime less than or equal to $\sqrt{n}$ divides $n$, then $n$ is a prime
43059
Show that $g^k=e$ if and only if the order of $g$ is finite and divides $k$.
157884
For any prime $p>3$ show that 3 divides $2p^2+1$
109778
Proving $(x-c)|(p(x)-p(c))$
3786
Help finding a combinatorial proof of $k {n \choose k } = n {n - 1 \choose k -1}\;$
88656
For $p$ prime, show that $\Phi_{p^n}(x) = 1 + x^{p^{n-1}} + x^{2p^{n-2}} + \dots +x^{(p-1)^{p^n-1}}$
155886
let q be an odd prime and p a prime dividing $1+n+\dots+n^{q-1}$
100984
Prove that if prime $p$ divide $a_{2k}-2$, then $p$ divide also $a_{2k+1}-1$.
68145
If $p$ is an odd prime and $k$ an integer with $0<k<p-1$ then $1^k + 2^k + \ldots + (p-1)^k$ is divisible by $p$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
12365
Show that every prime $p>3$ is either of the form $6n+1$ or of the form $6n+5$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
131488
Show that $p^3+4$ is prime
149792
Show that $\lim_{p \to 0} \mathbb {E} [(X^p-1)/p ] = \mathbb{E} [\log X]$
115726
For $p,q$ prime and $p \neq q$ show that $C_p \times C_q \cong C_{pq}$
110876
Prove that $p\mid \binom{p}{k},\ 0< k< p$