If $R$ is reduced residue system then $\exists! b\in R$ such that $a\equiv b\pmod n$
190
Why $\gcd(qb+r,b)=\gcd(b,r)$?
121717
Prove that $z^2 \equiv ab\pmod p$ is solvable if and only if both or neither of $x^2 \equiv a \pmod p$, $x^2 \equiv b \pmod p$ are solvable.
113248
Proving $r!$ divides the product of r succesive positive integers
148455
Show that if $(a,n)=d>1$ and $k$ is any positive integer, then $a^k\not\equiv 1\pmod{n}.$
116585
$r$ primitive root of prime $p$, where $p \equiv 1 \mod 4$: prove $-r$ is also a primitive root
134450
Chinese Remainder Theorem $\ x\equiv m_k-2\pmod{m_k}$
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 $b^2\equiv a \mod p$.
39800
Existence of solution for a linear system mod 2
72672
Show that if $ar + bs = 1$ for some $r$ and $s$ then $a$ and $b$ are relatively prime
64453
Prove if $(a,p)=1$, then $\{a,2a,3a,...,pa\}$ is a complete residue system modulo $p$.
84746
elementary proof that infinite primes quadratic residue modulo $p$
78356
How to prove ${n \choose p} \equiv\left\lfloor \frac{n}{p} \right\rfloor\pmod p$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
81346
Let $r$ be the smallest positive integer such that $a^r \equiv 1 \bmod{n}$. Prove $r \mid \phi(n)$.
111505
Proof $\binom{p-1}{k}\equiv (-1)^k \mod{p}$
29979
When is $-3$ a quadratic residue mod $p$?