Proving that $\gcd(ac,bc)=|c|\gcd(a,b)$
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
23066
If $\gcd(a, b) = 1$, then $\gcd(ab, c) = \gcd(a, c) \cdot\gcd(b, c)$
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
114780
$\forall a,b,c \in \mathbb{N}\!:\ a\mid b \Rightarrow ac\mid bc. $ Divisibility Proof
157508
prove $lcm(a,gcd(b,c))=gcd(lcm(a,b),lcm(a,c))$
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
11600
$\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$
142693
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
82777
If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
55441
Check: For the integers $a,b,c$ show that $\gcd(a,bc)=\gcd(a,\gcd(a,b)\cdot c)$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
11168
Proof of divisibility: if $a|b$ and $a|(b+c)$ then $a|c$
107914
Expanding GCDs Reasoning: $\gcd(a,c) \cdot \gcd(b,c) = \gcd(ab,bc,ac,cc)$
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$