Showing $\gcd(a,b,c) = \gcd(a,\gcd(b,c))$ without using Bezout's lemma and only the definition of the gcd
2208
$\gcd(a,b,c)=\gcd(\gcd(a,b),c)\,$ [GCD Associative Law]
73391
Prove that if $\gcd(a,b)=1$, then $\gcd(a^2,b^2)=1$
45213
Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
94914
Concise proof that every common divisor divides GCD without Bezout's identity?
20003
Prove for positive integers a,b,c and d (where b does not equal d), if gcd(a,b) = gcd(c,d) = 1, then a/b + c/d is not an integer
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
25573
If $\gcd(a,b)=1$, then $\gcd(a+b,a^2 -ab+b^2)=1$ or $3$.
74631
For natural numbers $a$ and $b$, show that $a \Bbb Z + b \Bbb Z = \gcd(a, b)\Bbb Z $
24160
Show that if $a$ is an odd integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
154959
Proof of associativity of GCD?
7091
If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
57281
Is Bezout's lemma enough to confirm the GCD of a number.
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
78031
$c\mid a,b\iff c\mid\gcd(a,b)$ [GCD Universal Property]