Prove that the system $x\equiv a \pmod m$, $x\equiv b \pmod n$ has a solution iff $\gcd(m,n) \mid (a-b)$
14658
Deriving Chinese Remainder Theorem from gcd Bezout identity
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
133925
Find All Solutions to System of Congruence
53915
prove that $a \equiv b \mod m$ is an equivalence relation on the integers
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
77164
If $a \mid m$ and $(a + 1) \mid m$, prove $a(a + 1) | m$.
132587
Prove that the product of coprimes must be congruent to $-1$ or $1 \pmod{m}$
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
32344
Let $a,k,m$ be integers. Prove that $\gcd(ka,km) = k\gcd(a,m)$.
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
128107
find all $ a \in \mathbb N $ such that $2^a − 1 \equiv 0 \pmod 3$ and $ 4x^2 + 1 \equiv 0 \pmod {\frac{2^a -1}{3}}$ has a solution
17601
How to prove that if $a\equiv b \pmod{kn}$ then $a^k\equiv b^k \pmod{k^2n}$
78852
If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
47232
Prove $\forall a,b,c \in \mathbb Z$, if $ab+ac\equiv 3\pmod 6$ then $b \not\equiv c \pmod 6$ by contraposition
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$