Congruence proof $ax \equiv b $ mod $m$
14809
Existence & uniqueness of solutions of linear congruences $\ cx\equiv b\pmod m$
47232
Prove $\forall a,b,c \in \mathbb Z$, if $ab+ac\equiv 3\pmod 6$ then $b \not\equiv c \pmod 6$ by contraposition
13317
show that $a^{12} \equiv 1 \mod 32760$ for each $a \in \mathbb{Z}$ such that $\text{gcd}(a, 32760) = 1$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
156966
Prove that if $b = dk$ where $gcd(m, a) = d$, there are $d$ solutions to $ax \equiv b \mod m$
26478
Show that the congruence $x^{p-1} \equiv 1 \pmod {p^2}$ has precisely $p-1$ solutions modulo $p^2$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
23830
Proof that if $p\equiv3\,\left(\mbox{mod 4}\right)$ then $p$ can't be written as a sum of two squares
63769
Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
90438
To prove that if $ab\equiv ac\bmod n$ and $(a,n)=1$ then $b\equiv c\bmod n$
101057
Show that $1^{p-1} + 2^{p-1} +\ldots + (p-1)^{p-1} \equiv -1 \mod p$
23153
$\bmod p\!:\ x^2\equiv 1\iff x\equiv \pm1\ $ for prime $p$
12636
Product Rule for mod operator: $ ab\bmod p = ((a\bmod p)(b\bmod p))\bmod p$
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
94017
If p is a prime and a ∈Z. How do we prove that the congruence x^x ≡ a (mod p) is solvable?
92518
mod Distributive Law, factoring $\!\!\bmod\!\!:$ $\ ab\bmod ac = a(b\bmod c)$