$\gcd(m^2,n^2)$ = $(\gcd(m,n))^2$
34330
How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
92630
How to prove $\gcd(a^m,b^m) = \gcd(a,b)^m$ using Bézout's Lemma
7091
If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
14693
$d = \gcd(c,m),\ ac≡bc\pmod{\! m}\Rightarrow a≡b\pmod {\!m/d}\ $ [Congruence Cancellation & Division Rule]
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
107095
Prove $\gcd(a,b) \gcd(a,c) \gcd(b,c) \,\text{lcm} (a,b,c)^2=$ $\text{lcm}(a,b)\,\text{lcm}(a,c) \,\text{lcm}(b,c) \gcd(a,b,c)^2$
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
158025
Intuition of why $\gcd(a,b) = \gcd(b, a \pmod b)$?
76126
How to find the GCD of two polynomials
32344
Let $a,k,m$ be integers. Prove that $\gcd(ka,km) = k\gcd(a,m)$.
1713
number theory $\gcd(a,bc)=\gcd(a,c)$
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
65687
$\gcd(a,b)\gcd(x,y)\mid ax+by$