How to prove that if $a^x \equiv a^y \pmod n$ then $x \equiv y \pmod{\operatorname{ord}_{n}(a)}$?
4972
Mod of numbers with large exponents [modular order reduction]
73238
Show that $a^{q-1} \equiv 1 \pmod{pq}$
20467
Show that $(y_n)\in l^q$
70463
Let $p$ be prime and $a$ be a natural not divisible by $p$. Then, $\exists$ integers $x,y$ such that $ax\equiv y \pmod{p}$ with $0<|x|,|y|<\sqrt{p}$
44033
Number theory lemma - $x^2 \equiv 1 \pmod p$ implies $x\equiv\pm 1\pmod p$
141369
If $a+b\equiv0\pmod m$ and $c+d\equiv 0\pmod m$ then $ac\equiv bd\pmod m$. Demonstration.
125209
Prove $e^x - e^y \leq e |x-y|$ for $x$ belonging to $[0,1]$
38636
A basic question about exponentiation: when is $a^{xy}=(a^x)^y$?
100426
Let $a$ and $b$ be integers. Show that if $a \equiv b \pmod{p}$ for every prime $p$, then $a = b$.
131212
Prove that if $A$ is regular then $\operatorname{adj}(\operatorname{adj}(A)) = (\det A)^{n-2} A$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
25419
If $n\geq m$ then $(x^m+y^m)^{1/m} \ge (x^n+y^n)^{1/n}$
145397
How to prove that for every $i$, $a_i = i$
112390
Solutions of $x^2 \equiv 1 \pmod N$
11989
$ac\equiv bc \pmod{\!m}\!\!\iff\!\! a\equiv b \pmod{\!\!\frac{m}{(c,m)}}$ [Euclid's Lemma, congruence form]
103732
Proving that for $n \equiv 0 \pmod{2}$, we get ${n \choose 0} < {n \choose 1} <\ldots< {n \choose n/2-1}<{n\choose n/2}$ etc.
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$