How can you prove the existence of an integer for ab congruent to 1 (mod p).
2904
Solving linear congruence (modular inverse or fraction) via gcd Bezout equation
6160
How to prove a number $a$ is quadratic residue modulo $n$?
71788
Prove existence of multiplicative inverse modulo p when p is prime
144851
For a prime $p \equiv 1$ or $3$ (mod $8$), show that the equation $x^{2} + 2y^{2} = p$ has a solution.
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
152039
Prove that $a^{2^n}=1 \mod 2^{n+2}$
45882
Solving a congruence without Fermat's little theorem
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 $b^2\equiv a \mod p$.
30630
Proving ${n \choose p} \equiv \Bigl[\frac{n}{p}\Bigr] \ (\text{mod} \ p)$
131241
primes congruent to 5 mod 6 and 1 mod p
46147
Proving $(\forall a\in\mathbb{Z^+})(m\in\mathbb{Z^+}\to a^m\equiv a^{m-\phi(m)}\pmod{m})$
44019
Show that for any prime $ p $, there are integers $ x $ and $ y $ such that $ p|(x^{2} + y^{2} + 1) $.
33735
The square of an integer is congruent to 0 or 1 mod 4
20161
Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
77806
Proving if $a$ is not a multiple of prime $p$, then an integer $b$ exists such that $p^b - 1$ is a multiple of $a$
108035
solving diophantine equations using congruences
14809
Existence & uniqueness of solutions of linear congruences $\ cx\equiv b\pmod m$