Show that if $(a,b)=1$, $a\mid c$ and $b\mid c$, then $a\cdot b\mid c$ 
Show that if $(a,b)=1$, $a\mid c$ and $b\mid c$, then $a\cdot b\mid c$.

Tried
$c=a\cdot k$ and $c=b\cdot j$ with $k,j\in\mathbb{N}$ then $a\cdot b\mid c^2=c\cdot c$.

You can generalize this theorem further:
If $a \mid c$ and $b \mid c$ then $\operatorname{lcm}(a,b) \mid c$ where $\operatorname{lcm}(a,b)$ is the least common multiple of $a$ and $b$. This is obvious by definition of lcm but you also need to prove that $\displaystyle \operatorname{lcm}(a,b) = \frac{|a \cdot b|}{\gcd(a,b)}$.
