If $k\mid n$ and $k\mid m$ then $k\mid\gcd(n,m)$
75039
Smallest positive element of $ \{ax + by: x,y \in \mathbb{Z}\}$ is $\gcd(a,b)$
90283
Proof that $3 \mid \left( a^2+b^2 \right)$ iff $3 \mid \gcd \left( a,b\right)$
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
85688
How to show $\gcd(ac, bd) = \gcd(a,d) * \gcd(b,c)$
97196
$k^{\phi(l)} + l^{\phi(k)} \equiv1\pmod{lk}\,$ if $\,\gcd(l,k)=1$
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
156966
Prove that if $b = dk$ where $gcd(m, a) = d$, there are $d$ solutions to $ax \equiv b \mod m$
18018
Proving that $4 \mid m - n$ is an equivalence relation on $\mathbb{Z}$
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power
157784
Prove that if $\gcd(ab,c)=1$, then $\gcd(a,c)=1$.
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
72640
$(m,n) = (\gcd(m,n))$ as ideals in $\Bbb Z$
102757
Prove that either $m$ divides $n$ or $n$ divides $m$ given that $\operatorname{lcm}(m,n) + \operatorname{gcd}(m,n) = m + n$?
37346
If $m, n$ be positive integers, prove that $\phi(mn)=\phi((m,n))\phi([m,n])$, where $(m,n)=$ gcd of $m, n$ and $[m, n]=$ lcm of $m, n$.
153100
A counterexample to "$k\mid n$ if and only if $k\mid n^2$"