Prove $\gcd(a,m) \mid \gcd(ab,m)$ $\forall a,b,m \in \Bbb Z$
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
37808
Show that $\gcd(a^2, b^2) = \gcd(a,b)^2$
19656
Proving that $ \gcd(a,b) = as + bt $, i.e., $ \gcd $ is a linear combination.
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
34330
How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
30752
Proof of $(\mathbb{Z}/m\mathbb{Z}) \otimes_\mathbb{Z} (\mathbb{Z} / n \mathbb{Z}) \cong \mathbb{Z}/ \gcd(m,n)\mathbb{Z}$
45213
Easiest and most complex proof of $\gcd (a,b) \times \operatorname{lcm} (a,b) =ab.$
114667
$d\mid a,b \iff d\mid\gcd(a,b) \ $ [GCD Universal Property]
105832
prove that if $\gcd(a,b)=1$ then $\gcd(a^2, b^2)=1$
22845
Prove that if d is a common divisor of a and b, then $d=\gcd(a,b)$ if and only if $\gcd(a/d,b/d)=1$
55673
Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
81794
Prove that $(ab,cd)=(a,c)(b,d)\left(\frac{a}{(a,c)},\frac{d}{(b,d)}\right)\left(\frac{c}{(a,c)},\frac{b}{(b,d)}\right)$
121156
Prove that: $\gcd[a,b,c]=\frac{abc.\operatorname{lcm}(a,b,c)}{\operatorname{lcm}(a,b)\operatorname{lcm}(a,c)\operatorname{lcm}(b,c)}$
24160
Show that if $a$ is an odd integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$
72747
Proving gcd($a,b$)lcm($a,b$) = $|ab|$
19700
Prove that if $a$ and $b$ are relatively prime, then $\gcd(a+b, a-b) = 1$ or $2$