If $\gcd(x;y)=1$, why is $\gcd[(ax+by);(cx+dy)]=1$?
25825
Proof that $\gcd(ax+by,cx+dy)=\gcd(x,y)$ if $ad-bc= \pm 1$
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
38295
If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?
142693
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$
7151
Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
25713
If $a$ and $b$ commute and $\text{gcd}\left(\text{ord}(a),\text{ord}(b)\right)=1$, then $\text{ord}(ab)=\text{ord}(a)\text{ord}(b)$.
108450
Prove that $\gcd(a, b) = \gcd(a, b + ma)$?
151434
If integers $a$ and $b$ are not both zero, then $\gcd(a,b)=\gcd(a-b,b)$
1713
number theory $\gcd(a,bc)=\gcd(a,c)$
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
128199
Proving gcd(a, a+2) is 1 or 2 for every integer a?
4486
If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$
59856
Find $x,y$ given $\gcd(x,y)$ and ${\rm lcm}(x,y)$
101394
Please explain the logic behind $d(xy) = y(dx) + x(dy)$
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
5789
$ d = \gcd(a,b)\Rightarrow\ \gcd(a/d,b/d) = 1$
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.