Let $a, b$ and $m$ be integers, and $m \ge 2, \ ab ≡ [ (a \bmod m) * (b \bmod m) ] (\bmod m).$
16548
Prove $(x+y) \text{ mod } n = ((x \text{ mod } n)+(y \text{ mod } n)) \text{ mod } n,\,$ and product analog
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
6332
If $p\equiv1\mod{4}$, then $p=a^{2}+b^{2}$ where a and b are uniquely determined
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
110703
How Can I prove (a+b) mod m = (a mod m) + (b mod m)) mod m?
63931
When $ab/(a+b)$ is an integer, where $a,b$ are positive integers.
52530
There exists a prime congruent to $m$ mod $p$
124995
$p\equiv 1,2,3\text{ mod }8 \Rightarrow p=a^2+2b^2\text{ for some $a,b\in\mathbb{Z}$.}$
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
32344
Let $a,k,m$ be integers. Prove that $\gcd(ka,km) = k\gcd(a,m)$.
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
6410
Prove that $(ma, mb) = |m|(a, b)\ $ [GCD & LCM Distributive Law]
13026
Prove that if AB=I_m, then rk(A)=rk(B).
111555
Find $a,b \in \mathbb{N}$ with $\mathrm{lcm}(a,b)=12\gcd(a,b)\,$ and $\,a\bmod b = 5$
28021
Showing that these two definitions of $\gcd(a,b)$ are equivalent
46286
Prove if $a\mid b$ and $b\mid a$, then $|a|=|b|$ , $a, b$ are integers.