Let $a^n, a^m \in (a^k)$ for some positive integer $k$. Then $k \mid n, m.$ Hence $k \mid \operatorname{gcd(n, m)}?$
78031
$c\mid a,b\iff c\mid\gcd(a,b)$ [GCD Universal Property]
97469
How to prove $|a^k|=n/\gcd(n,k)$ whenever $|a|=n$?
11600
$\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$
59533
If $\gcd(m,n)=1$, then $\mathbb{Z}_n \times \mathbb{Z}_m$ is cyclic.
63547
Euclid's Lemma and $\gcd(a,b)=1\Rightarrow \gcd(a^n,b^k)= 1$
51858
Showing $\gcd(2^m-1,2^n+1)=1$
7843
Show that order of $a^k$
24572
Does $n \mid 2^{2^n+1}+1$ imply $n \mid 2^{2^{2^n+1}+1}+1$?
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
39217
If $d \mid n$ then $x^{q^d} - x \mid x^{q^n} - x$.
96535
Show that if $\gcd(a,3)=1$ then $a^7 \equiv a\pmod{63}$. Why is this assumption necessary?
31465
Prove that if $a \mid b$ then $\gcd(a,b) = |a|$
147902
Prove $a \mid m$ and $b \mid m \implies \frac{a \cdot b}{\text{gcd}(a, b)} \mid m$
101872
Prove that $\gcd(a^2, b^2) = \gcd(a, b)^2$
5708
Prove that if $k\ge 1\,$ and $\,a^k\equiv 1\pmod{\! n}$ then $\gcd(a,n)=1$
7967
Prove that the order of the cyclic subgroup $\langle g^k\rangle $ is $n/{\operatorname{gcd}(n,k)}$
112031
Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.