Let $a,b\in\Bbb{N}$ such that $\gcd(a,b)=1$ and $n,r\in\Bbb{N}^*$. If $ab=n^r$ then there exists $k,l\in\Bbb{N}$ such that $a=k^r$ and $b=l^r$.
54554
If a product of relatively prime integers is an $n$th power, then each is an $n$th power
1315
Let $R$ be a ring. Define a circle composition ◦ in R by $a ◦ b =a+b-ab$, $a, b ∈ R$.
66748
Show that if $a \equiv b \pmod n$, $\gcd(a,n)=\gcd(b,n)$
91506
If $\gcd(a,b) = 1$ then $\gcd(a,a + b) = 1$
24845
Let $a\mid c$ and $b\mid c$ such that $\gcd(a,b)=1$, Show that $ab\mid c$
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
78678
If $\gcd(m,n)=1$, and $\gcd(a,m)=\gcd(b,n)=1$, then $\gcd(ab, mn)=1$
7151
Let $a,m,n \in \mathbf{N}$. Show that if $\gcd(m,n)=1$, then $\gcd(a,mn)=\gcd(a,m)\cdot\gcd(a,n)$.
25907
Show $a\Bbb Z+b\Bbb Z = \gcd(a,b)\Bbb Z$
147527
Let $a$ and $b$ be non-zero integers, and $c$ be an integer. Let $d = \gcd(a, b)$. Prove that if $a|c$ and $b|c$ then $ab|cd$.
55673
Alternate proof that $\exists\ x \in \Bbb{Z}$ such that $ \gcd (a+bx,c) = 1$?
32344
Let $a,k,m$ be integers. Prove that $\gcd(ka,km) = k\gcd(a,m)$.
102133
Prove: $\gcd(a,b) = \gcd(a, b + at)$.
72747
Proving gcd($a,b$)lcm($a,b$) = $|ab|$
126358
Prove for $\gcd$ of two integers $a,b$, that $(a^n,b^n) = (a,b)^n$
156966
Prove that if $b = dk$ where $gcd(m, a) = d$, there are $d$ solutions to $ax \equiv b \mod m$
69967
How to prove $\gcd(a^m-b^m,a^n-b^n) = a^{\gcd(m,n)} - b^{\gcd(m,n)} $?