Let $n=(6t+1)(12t+1)(18t+1)$. Show that $a^{n-1} \equiv 1\pmod n$ if $(a,n)=1$
44063
Suppose k is an integer such that $6k + 1$, $12k + 1$, $18k + 1$ are primes. Show that $n = (6k + 1)(12k + 1)(18k + 1)$ is a Carmichael number.
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
2218
Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime [Wilson's Theorem]
96535
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
102334
How can you prove $\frac{n(n+1)(2n+1)}{6}+(n+1)^2= \frac{(n+1)(n+2)(2n+3)}{6}$ without much effort?
63183
Suppose $a,b,m,n$ are natural numbers, prove that if $a({m \over n})=b$ and $(a,n)=1$, then ${m \over n} \in \mathbb{N}$
101057
Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
40274
If $|a| = 12, |b| = 22$ and $\langle a \rangle\cap \langle b\rangle \ne e$, prove that $a^6 = b^{11}$
60305
Proving the congruence $p^{q-1}+q^{p-1} \equiv 1 \pmod{pq}$
32166
Prove: if $n\mid 7^n+6^n$ and $n>1$, then $13\mid n$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
81058
Evaluate $6^{433} \pmod {21}$ and a proving question
76696
Using induction to prove $2^{n-1}(1 + a_1a_2\ldots a_n) \geq (1+a_1)(1+a_2)\ldots(1+a_n)$ for $a_i \geq 1$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
89175
Proof of $(a+b)^{n+1}$
100426
Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
102860
solve $3x^2 + 6x +1 \equiv 0 \pmod {19}$