Is this a valid proof for $\gcd(a^m - b^m, a^n - b^n) = a^{\gcd(m, n)} - b^{\gcd(m,n)}$?
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
28021
Showing that these two definitions of $\gcd(a,b)$ are equivalent
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
69279
If $\gcd(a,b)=1$ then, $\gcd(a^2,b^2)=1$
19500
How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
60842
if $a^2+b^2=c^2$ then $gcd(a,b)=gcd(a,c)=gcd(b,c)$.
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
10105
Proving $\, a^b = (a\bmod n)^b\, \pmod{\!n}\ $ [Congruence Power Rule]
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
108450
Prove that $\gcd(a, b) = \gcd(a, b + ma)$?
59006
Proof of $\gcd(n!+1, (n+1)!+1)=1$?
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$