Proof that if $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$.
20878
If $(a,b)=1$ then prove $(a+b, ab)=1$.
55673
Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
7316
How to prove that $z\gcd(a,b)=\gcd(za,zb)$
66785
Find $\gcd(p^n-1,p^m+1)$.
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?
7091
If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
128820
Prove that if $\gcd(a,b)=1$ then $\gcd(ab,c) = \gcd(a,c) \gcd(b,c)$
106416
If $n,m \in \mathbb{N}$ then there are $c,d$ such that $cd = (m,n)$, $(c,d) = 1$ and $(m/c,n/d) = 1$.
81864
Finite group of order $mn$ with $\gcd(m,n) = 1$ .
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
6012
Prove that $\gcd(a^n - 1, a^m - 1) = a^{\gcd(n, m)} - 1$
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
2289
Prove that if $\gcd( a, b ) = 1$ then $\gcd( ac, b ) = \gcd( c, b ) $
10348
Prove that if the $\gcd(a,b)=1$, then $\gcd(a+b,ab)=1$
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
52483
Showing that $(m)\cap (n)=(\operatorname{lcm}(m,n))$ and $(m)+(n)=(\gcd(m,n))$ for any $m,n\in\mathbb{Z}$