How can I show that in a UFD $\operatorname{gcd} (ra,rb) \sim r \operatorname{gcd} (a,b)$?
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
116175
Show that $\gcd(3k+2,5k+3)=1$
147802
Proving that $\gcd\left(\frac a {\gcd(a,b)},\frac b {\gcd(a,b)}\right) =1$
142693
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
83871
GCDs in integral domains are unique up to unit factors (associates)
11509
Relationship between $\operatorname{ord}(ab), \operatorname{ord}(a)$, and $\operatorname{ord}(b)$
65687
$\gcd(a,b)\gcd(x,y)\mid ax+by$
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
8160
What is $\gcd(0,a)$, where $a$ is a positive integer?
49294
prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
114667
$d\mid a,b \iff d\mid\gcd(a,b) \ $ [GCD Universal Property]
117688
Prove that $\def\Aut{\operatorname{Aut}}\Aut(\mathbf{Z_{n}})\simeq \mathbf{Z_{n}^{*}}$
74187
$\operatorname{ord}(u)=r,\,\operatorname{ord}(v)=s$ $\,\Rightarrow\,\operatorname{ord}(uv)=rs\,$ if $\,r,s\,$ (co)primes
2208
$\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
107095
Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$