Suppose that $a \equiv b\mod m$ and that $a \equiv b\mod n $ .Assuming gcd$(m;n) = 1 $ ,prove that $a \equiv b\mod mn$
21250
Prove $x \equiv a \pmod{p}$ and $x \equiv a \pmod{q}$ then $x \equiv a\pmod{pq}$
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
63769
Group theory proof of Euler's theorem ($a^{\phi(m)} \equiv 1\mbox{ }(\mbox{mod }m)$ if $\gcd(a,m)=1$)
152299
Proving $a^{n-1} \equiv 1 \mod n $ when n is not prime.
95889
Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
102757
Prove that either $m$ divides $n$ or $n$ divides $m$ given that $\operatorname{lcm}(m,n) + \operatorname{gcd}(m,n) = m + n$?
7090
Show that $\gcd(a,bc)=1$ if and only if $\gcd(a,b)=1$ and $\gcd(a,c)=1$
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
126358
Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
22845
Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
84733
Why $a\equiv b\pmod n$ implies $a$ and $b$ have equal remainder when divided by $n$?
32463
$ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
34323
Prove that $a^{(p-1)/2} \equiv 1$ (mod p) and $a^{(p-1)/2} \equiv -1 \pmod p$
125868
Prove $a \equiv b \pmod{c} \implies a^n \equiv b^n \pmod{c}$.