An integer $k$ is a generator of group $\mathbb{Z_n}$ if and only if $\gcd(k, n)=1$
67004
Power of a generator generates the group iff this power is coprime to the group order.
78852
If $\gcd(m,n) =1, $ then $m\mathbb{Z}/ mn\mathbb{Z} \cong \mathbb{Z}/(n)$
153100
A counterexample to "$k\mid n$ if and only if $k\mid n^2$"
67926
$S_n$ is generated by elements of the form $(1k)$
147079
$\gcd\left(\binom{n}{k},n\right)>1$ for nontrivial values?
87013
If $\gcd(a,b)=1$, then there exists integers $x$ and $y$ such that $xa + yb = 1$
117178
Cyclic group generators
147517
Any group of order $n$ satisfying $\gcd (n, \varphi(n)) =1$ is cyclic
151434
If integers $a$ and $b$ are not both zero, then $\gcd(a,b)=\gcd(a-b,b)$
143543
Prove that $(n - 1)^2 \mid n^k -1$ if and only if $(n - 1) \mid k$
92583
Prove that $N(\gamma) = 1$ if, and only if, $\gamma$ is a unit in the ring $\mathbb{Z}[\sqrt{n}]$
76504
Non-trivial cyclic group with unique generator
2043
$X^n-Y^m$ is irreducible in $\Bbb{C}[X,Y]$ iff $\gcd(n,m)=1$
59566
Proving that $ka \equiv kb \pmod{m} \iff a \equiv b \pmod{m/\gcd(k,m)}$
114187
In a finite cyclic group, the generator has the order of the group
66510
A group is abelian if and only if the center of the group is all the group
114221
A Group $G$ is soluble $\iff G^{(k)}=\{e\}$ for some $k$