$\gcd(a',b')=1\implies c=ka',d=kb'$
52672
Solution of $\frac{a}{b}=\frac{a'}{b'}$ if $a,b,a',b' \in \mathbb{N}$
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
28021
Showing that these two definitions of $\gcd(a,b)$ are equivalent
63758
$c\mid ab\Rightarrow c\mid \gcd(a,c)\gcd(b,c),\,$ for integers $a, b, c$
60842
if $a^2+b^2=c^2$ then $gcd(a,b)=gcd(a,c)=gcd(b,c)$.
25825
Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
84417
Prove that if $\gcd(a,b)=1$ then $\gcd(a^m, b)=1$
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
65687
$\gcd(a,b)\gcd(x,y)\mid ax+by$
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
1991
If $\gcd(a,b)=1$ and $a$ and $b$ divide $c$, then so does $ab$
4486
If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$
142693
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$