Is this a valid proof for $\gcd(a,b) = \gcd(b,r),\ r = a-bq$?
190
Why $\gcd(qb+r,b)=\gcd(b,r)$?
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
76507
$\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$
85688
How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
82777
If $\gcd(a, b) = 1$ then prove that $\gcd(a+b, a^2-ab+b^2) = 1$ or $3$?
117098
Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
82938
$\gcd(a,b)=1, x^a = y^b\Rightarrow x = n^b$, $ y = n^a$ for an integer $n$.
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
151434
If integers $a$ and $b$ are not both zero, then $\gcd(a,b)=\gcd(a-b,b)$
149993
Use the Euclidean algorithm to prove that gcd(na, nb) = n gcd(a, b).
34738
Prove that $\gcd(n!+1,(n+1)!+1)=1$
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
158025
Intuition of why $\gcd(a,b) = \gcd(b, a \pmod b)$?
108450
Prove that $\gcd(a, b) = \gcd(a, b + ma)$?
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
4486
If $ar + bs =1$, then $\gcd(a,s) = \gcd(r,b) = \gcd(r,s) = 1$