Prove that in ring $\mathbb{Z}, \forall a,b \in \mathbb{Z} $, $(a,b) = (d)$, where $ d = \mathrm{gcd}(a,b)$
72640
$(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
32344
Let $a,k,m$ be integers. Prove that $\gcd(ka,km) = k\gcd(a,m)$.
56612
Proof of $\gcd(a,b)=ax+by\ $ [Bezout's identity]
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
8787
In a ring $(A,+, \cdot)$ if $aba = a$ then $bab = b$ and all non zero elements in $A$ are invertible.
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
142693
$\gcd(a,b) = \gcd(a + b, \mathrm{lcm}[a,b])$
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
117098
Properties of the greatest common divisor: $\gcd(a, b) = \gcd(a, b-a)$ and $\gcd(a, b) = \gcd(a, b \text{ mod } a)$
32108
$ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
21788
Proving that if $ad-bc = \pm 1$, then $\gcd(x,y) = \gcd(ax +by, cx + dy)$
147527
Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.