Show that there is exactly one element in $A$ congruent to $1$ modulo $p$ as follows.
380
Bijection from modular multiplication?
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
102234
Show $a^p \equiv a \mod 2p$
19071
solution set for congruence $x^2 \equiv 1 \mod m$
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
73020
Every integer is congruent to the sum of its digits mod 9
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
46803
Proof that there are infinitely many primes congruent to 3 modulo 4
86977
If $\cup \mathcal{F}=A$ then $A \in \mathcal{F}$. Prove that $A$ has exactly one element.
135465
Group theory proof of existence of a solution to $x^2\equiv -1\pmod p$ iff $p\equiv 1 \pmod 4$
96062
Show $\underbrace{{111\cdots}1}_{{\small{p-1} \ 1's}}$ is divisible by $p$
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
32463
$ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
84746
elementary proof that infinite primes quadratic residue modulo $p$
111490
Proof that every number has at least one prime factor
45882
Solving a congruence without Fermat's little theorem
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$