Problem:
Let $a$ and $b$ be nonzero elements of euclidean domain $A$ and let $p_1,...,p_s$ be a maximal system of irreducible divisors, pairwise non-associated, of $ab$.
Writing $a=up_1^{e^1}...p_{_s}^{e^s}$ and $b=v_{p_1}^{f^1}...p_{_s}^{f^s}$ for $u,v \in A^x$
I wish to show that the $lcm(a,b)$ ~ $p_{_1}^{h^1}...p_{_s}^{h^s}$ (they are associates) where $h_i=max(e_i,f_i)$ for $1\leq i \leq s$.
My working definition of the $lcm$ say $m$ of {$a_1,...,a_s$} in a euclidean domain is:
$i)$ $m$ is nonzero and {$a_1,...,a_s$} each divide $m$
$ii)$ if a nonzero element $c \in A$ is divisible by {$a_1,...,a_s$} then $m|c$.
Thoughts:
I am having difficulty using the defintion to show that $p_{_1}^{h^1}...p_{_s}^{h^s}$|$m$. Any hints appreciated.