For Fibonacci numbers: $\gcd(F_m, F_n) = F_{\gcd(m, n)}$
14210
Fibonacci divisibilty properties $ F_n\mid F_{kn},\,$ $\, \gcd(F_n,F_m) = F_{\gcd(n,m)}$
14693
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
28021
Showing that these two definitions of $\gcd(a,b)$ are equivalent
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
92630
How to prove $\gcd(a^m,b^m) = \gcd(a,b)^m$ using Bézout's Lemma
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
76507
$\gcd(b^x - 1, b^y - 1, b^ z- 1,\dots) = b^{\gcd(x, y, z,\dots)} -1$
67660
Prove $\,b\equiv c\pmod{\!m}\Rightarrow \gcd(b,m)=\gcd(c,m)$ i.e. $\gcd(c+km,m) = \gcd(c,m)$
48610
Fibonacci Numbers Proof: $ f_n = \binom n0 + \binom{n-1}1 +\dots+ \binom{n-k}k$
38404
Prove $\gcd(a+b, a-b) = 1$ or $2\,$ if $\,\gcd(a,b) = 1$
133150
Fibonacci combinatorial identity: $F_{2n} = {n \choose 0} F_0 + {n\choose 1} F_1 + ... {n\choose n} F_n$
112948
Fibonacci addition law $F_{n+m} = F_{n-1}F_m + F_n F_{m+1}$
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
48839
Show that the order of $a\times b$ is equal to $nm$ if gcd(n,m)=1
122096
Find $GCD(n^2+1,n+1)$
19164
On the GCD of a Pair of Fermat Numbers