Understanding proof of $a\equiv b\pmod{n}\implies r_n(a)=r_n(b)$
30725
Why are naturals < modulus equal if congruent? [uniqueness of remainder]
158025
Intuition of why $\gcd(a,b) = \gcd(b, a \pmod b)$?
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
18018
Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$
38876
$r! \equiv (−1)^k \pmod p$
101159
Prove If a ≡ b (mod n) and c ≡ d (mod n), then a + c ≡ b + d (mod n)
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
13026
Prove that if AB=I_m, then rk(A)=rk(B).
12210
Show that $a^{61} \equiv a\pmod{1001}$ for every $a \in \mathbb{N}$
135986
Show if $a \equiv b \pmod{m}$ then $ac \equiv bc\pmod{mc}$
42780
Show that if $a^{n-1} \equiv 1 \pmod{n}$ for all $a$ such that $\gcd(n,a) = 1$, then $a^{n} \equiv a \pmod{n}$ for all $a$.
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
130342
Show that for any integer a and prime p, $(a+1)^p \equiv a^p+ 1 \pmod{p}$.
115866
Proof by induction; $a^n$ divides $b^n$ implies $a$ divides $b$
63879
Prove that $\sum_{k=0}^r {m \choose k} {n \choose r-k} = {m+n \choose r}$
107725
Prove that if $n$ is not divisible by $5$, then $n^4 \equiv 1 \pmod{5}$
29204
$a\equiv\bar a\!\pmod{\!kn}\Rightarrow a\equiv\bar a\!\pmod{\! n};\ $ $(a\bmod kn)\bmod n=a\bmod n.\ $ Congruences persist mod factors of the modulus