For any $a \in \mathbb{N}$, $4^a \equiv (3a+1) \mod 9$
127833
Proving that $7^n(3n+1)-1$ is divisible by 9
10105
Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule]
136367
Prove $ ∀a, b, c ∈ Z,$ if $ab + ac ≡ 3$ (mod 6) then b $\not\equiv$ c (mod 6).
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
96237
$1^2 + 2^2 + 3^2 + \cdots + (n-1)^2 \equiv 0$ mod n iff $n \equiv \pm 1$ mod 6
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
124240
Inductive proof of $\,9 \mathrel| 4^n+6n-1\,$ for all $\,n\in\Bbb N$
71451
A proof by strong induction that $a_n\le3^n$ where $a_n=a_{n-1}+a_{n-2}+a_{n-3}$
140198
Let $a$ be a quadratic residue modulo $p$. Prove that the number $b\equiv a^\frac{p+1}{4} \mod p$ has the property that $b^2\equiv a \mod p$.
59442
Are there $a,b>1$ with $a^4\equiv 1 \pmod{b^2}$ and $b^4\equiv1 \pmod{a^2}$?
95653
Euler's theorem: [3]^2014^2014 mod 98
27147
Proving $n! > n^3$ for all $n > a$
116585
$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
52951
Find all $n \in \mathbb{Z^+}$ such that $\phi(n)=4$
6332
If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
66861
Show that $11^{10} \equiv 1 (\mod 100)$
144067
calculating $a^b \!\mod c$