For $p\geq5, 1^2+2^2+3^2+...+(p-1)^2$ is divisible by $p$.
1837
Showing that $1^k+2^k + \dots + n^k$ is divisible by $n(n+1)\over 2$
85042
Write $1/1 + 1/2 + ...1/ (p-1)=a/b$ with $(a,b)=1$. Show that $p^2 \mid a$ if $p\geq 5$ is prime [Wolstenholme's theorem]
30620
For which natural n is $3^n+5^n$ divisible by $3^{n-1}+5^{n-1}$?
105442
Show that any prime divisor of $x^4+x^3+x^2+x+1$, with $x\in\mathbb{N}$, is $5$ or $1$ mod $5$
75733
Divisibility of prime numbers
14480
Proving $(P(AB))^2+(P(AB^c))^2+(P(A^cB))^2+(P(A^cB^c))^2\ge\frac{1}{4}$ for any events $A$ and $B$
13478
Prove that if $a^p-b^p$ is divisible by $p$, then it is also divisible by $p^2$
58182
How to prove that $\mathbb{Q}[\sqrt{p_1}, \sqrt{p_2}, \ldots,\sqrt{p_n} ] = \mathbb{Q}[\sqrt{p_1}+ \sqrt{p_2}+\cdots + \sqrt{p_n}]$, for $p_i$ prime?
137808
$n^2 + 3n +5$ is not divisible by $121$
29874
Not divisible by $2,3$ or $5$ but divisible by $7$
280
Prove by induction that $5^n - 1$ is divisible by $4$.
106847
Proof that $n^3+2n$ is divisible by $3$
52032
Prove that ${n^5 - n}$ is divisible by 5
26336
Prove that $n(n^2 - 1)(n + 2)$ is divisible by $4$ for any integer $n$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
6561
Prove that the expression $5^{2n+1} \cdot 2^{n+2} + 3^{n+2} \cdot 2^{2n+1}$ is divisible by 19.
3109
Prove that $(p+q)^m \leq p^m+q^m$