If $\gcd(u,m)=1$ then there exists an integer $k_1\in\Bbb Z$ such that $\gcd(mk_1+u,mk)=1$
36
Prove: If $\gcd(a,b,c)=1$ then there exists $z$ such that $\gcd(az+b,c) = 1$
32463
$ax\equiv b\pmod{\! n}$ has a unique solution if $\gcd(a,n) = 1$
138058
Prove that $\mathbb Z_{m}\times\mathbb Z_{n} \cong \mathbb Z_{mn}$ implies $\gcd(m,n)=1$.
48839
Show that the order of $a\times b$ is equal to $nm$ if gcd(n,m)=1
37596
If $\gcd(a,b)= 1$ and $a$ divides $bc$ then $a$ divides $c\ $ [Euclid's Lemma]
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
38295
If $\gcd (a,b) = 1$, what can be said about $\gcd (a+b,a-b)$?
78031
$c\mid a,b\iff c\mid\gcd(a,b)$ [GCD Universal Property]
66785
Find $\gcd(p^n-1,p^m+1)$.
19500
How can I find $\gcd(a^m+1,a^n+1)$ with $a,m,n$ positive integers?
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
18772
If $a,b \in\mathbb N$ and $\gcd(a,b)=1$, prove that $\gcd(a+b;a^2+b^2)= 1$ or $2$.
33996
Proving that $\gcd(2^m - 1, 2^n - 1) = 2^{\gcd(m,n )} - 1$
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
32108
$ a\in \Bbb Z_n$ is invertible $\,\Rightarrow\gcd (a,n) =1$
136064
How can I prove that $\gcd(a,b)=1\implies \gcd(a^2,b^2)=1$ without using prime decomposition?
31331
$|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power