Without using prime factorization, show if $m\mid n^2$ then $\gcd(m,n^2/m)\mid n$
34330
How to prove $\gcd(a^2,b^2) = (\gcd(a,b))^2$?
49294
prove that $\operatorname{lcm}(n,m) = nm/\gcd(n,m)$
138894
Prove that if $3\mid(a^2+b^2)$,then $3\mid a$ and $ 3\mid b$
97947
How can we prove that a binomial coefficient $n\choose m$is divisible by the ratio of $n$ and $\gcd(n,m)$?
40848
$a\mid b,c\mid d\Rightarrow\,\gcd(a,c)\mid \gcd(b,d)$
135779
Prove the following: If $a \mid bc$, then $a \mid \gcd(a, b)c$.
7091
If $\gcd(a,b)=1$, then $\gcd(a^n,b^n)=1$
36862
A question about proving that there is no greatest common divisor
92483
Without using prime factorization, find a prime factor of $\frac{(3^{41} -1)}{2}$
27544
With $a,\ b,\ m,\ n\in \mathbb{Z}$. Show that if ${\rm gcd}(m,n)=1$, then ${\rm gcd}(ma+nb,mn)={\rm gcd}(a,n){\rm gcd}(b,m)$
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
20823
If $\gcd(a,b) = 1$ and $a,b\mid x$ then $ab\mid x$.
14581
How do we prove $n^n \mid m^m \Rightarrow n \mid m$?
112031
Prove that $\gcd(2^{2^m}+1,2^{2^n}+1)=1$ if $m,n$ are positive integers.
61825
Congruence class $[a]$ modulo $m$, $\gcd(x, m) = \gcd(a, m)$
95889
Prove that $\gcd(n! + 1, 3n! + 1) = 1$, $\forall n \in \mathbb{N}$, $n \geq 2$.
65687
$\gcd(a,b)\gcd(x,y)\mid ax+by$