Proof that if $a \mid bc$ and $\gcd(a,b) = d$ then $a/d$ divides $c$
21241
Proving $\,a\mid bc\!\iff\! \frac{a}{(a,b)}\!\mid c\ $ [general Euclid's Lemma]
77064
Proving $\gcd((a+b)^m,(a-b)^m)\le2^m$ for coprime $a,b$ and $m \in \Bbb N$
113641
Suppose the gcd (a,b) = 1 and c divides a + b. Prove that gcd (a,c) = 1 = gcd (b,c)
35945
Proof that if $\gcd(a,b) = 1$ and $a\mid n$ and $b\mid n$, $ab \mid n$
77278
If $\,\gcd(a,b) = {\rm lcm}(a,b)\,$ then prove $a=b$
48452
Proof that $\gcd$ divides $\operatorname{lcm}$
153670
Prove that if $a \mid b$ and $b \mid c$, then $a \mid c$.
136854
Prove: For a,b,c positive integers, ac divides bc if and only if a divides b
106416
If $n,m \in \mathbb{N}$ then there are $c,d$ such that $cd = (m,n)$, $(c,d) = 1$ and $(m/c,n/d) = 1$.
130112
If $\gcd(a,n) = 1$ and $\gcd(b,n) = 1$, then $\gcd(ab,n) = 1$.
130653
$p^2$ divides $ab$ and $\gcd(a,b)=1$. Show $p^2$ divides $a$ or $p^2$ divides $b$
104068
Prove that $\forall a,b\in\mathbb{C} \exists c,d \in \mathbb{C}: c+d = a \land cd = b$
24160
Show that if $a$ is an odd integer and $b$ is an even integer then $\gcd(a,b)=\gcd(a,b/2)$
49109
Proving $\gcd \left(\frac{a}{\gcd (a,b)},\frac{b}{\gcd (a,b)}\right)=1$
80148
Prove $\gcd(a+b,a^2+b^2)$ is $1$ or $2$ if $\gcd(a,b) = 1$
114667
$d\mid a,b \iff d\mid\gcd(a,b) \ $ [GCD Universal Property]
147246
If $\gcd(a,c)=1=\gcd(b,c)$, then $\gcd(ab,c)=1$