Do we have that $x$ divides $cb$ with $x,b,c\in \mathbb{Z}$ implies $\gcd(x,b)>1$
33052
How to prove $\bar{m}$ is a zero divisor in $\mathbb{Z}_n$ if and only if $m,n$ are not coprime
57592
Does existence of $\gcd$ implies PID?
15168
If $a,b$ are positive integers such that $\gcd(a,b)=1$, then show that $\gcd(a+b, a-b)=1$ or $2$. and $\gcd(a^2+b^2, a^2-b^2)=1$ or $2 $
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
38295
If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?
5223
$\mathbb{Z}_m \times \mathbb Z_n$ is cyclic if and only if $\gcd(m,n)=1$
65218
Let $m \in \mathbb{Z^+} , n \in \mathbb{Z^+}$ and let $d=\gcd(m,n)$. Prove that $m\mathbb{Z}+n\mathbb{Z}=d\mathbb{Z}$
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
63758
$c\mid ab\Rightarrow c\mid \gcd(a,c)\gcd(b,c),\,$ for integers $a, b, c$
112345
In an Integral Domain is it true that $\gcd(ac,ab) = a\gcd(c,b)$?
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$