$\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$ I am trying to show that
$\gcd(a,\operatorname{lcm}(b,c))=\operatorname{lcm}(\gcd(a,b),\gcd(a,c))$.
If $d=\gcd(a,\operatorname{lcm}(b,c))$, then $d=ax-my$, where $m=\operatorname{lcm}(b,c)$. So
$d=ax-b(ky)$ and $d=ax-c(ly)$, which means that $\gcd(a,b)|d$ and $\gcd(a,c)|d$, or $d$ is a common multiple.
But I wasn't able to show that it is the smallest common multiple.
 A: Since you say you cannot use the simple min/max exponents proof via unique factorization, here is a proof that uses only universal gcd laws (so will work in any gcd domain). We simply eliminate all lcms by $\rm\,[x,y] = xy/(x,y),\:$ and apply gcd laws (distributive, commutative, associative, etc).
$$\rm\begin{eqnarray}
\rm &\rm\qquad\qquad (a,[b,c])\ &=&\rm\ [(a,b),(a,c)] \\
\rm \iff&\rm\qquad\quad \left(a,\dfrac{bc}{(b,c)}\right)\ & =&\rm\ \dfrac{(a,b)(a,c)}{(a,b,c)} \\
\iff &\rm (a,b,c)(a(b,c),bc)\ &=&\rm\ (a,b)(a,c)(b,c)
\end{eqnarray}$$
which is true since both sides $\rm = (aab,aac,abb,abc,acc,bbc,bcc)\:$ by distributivity etc.
If you are not proficient with  gcd laws, you may find it helpful to rewrite the proof employing a more suggestive arithmetical notation, namely denoting the gcd $\rm (a,b)\:$ by $\rm\ a \dot+ b.\:$ Because the arithmetic of GCDs shares many of the same basic laws of the arithmetic of integers, the proof becomes much more intuitive using a notation  highlighting this common arithmetical structure. Below is a sample calculation comparing the two notations.
$$\rm\begin{eqnarray}
\rm(a,\:b)\ (a,\:c) &=&\rm (a(a,\!\:c),b(a,\!\:c)) &=&\rm ((aa,ac),\:(ba,bc)) &=&\rm (aa,ac,\:\!ba,\:bc) \\
\rm\ (a\dot+ b)(a\dot+c) &=&\rm \color{#c00}{a(a\dot+c)}\dot+b(a\dot+c) &=&\rm (\color{#c00}{aa\dot+ac})\dot+(ba\dot+bc) &=&\rm aa\dot+ac\dot+ba\dot+bc
\end{eqnarray}$$
Now the gcd arithmetic looks like integer arithmetic, e.g. uses of the gcd distributive law look the same as for integers, e.g. $\ \rm \color{#c00}{a(a\!+\!c) = aa\!+\!ac}\,$ etc.
This property - that gcd distributes over lcm - plays a fundamental role in number theory, e.g. in the CRT solvability criterion for a system of congruencers
A: See if this post from PhysicsForums helps at all, the two problems seem somewhat interchangeable. 
http://www.physicsforums.com/showthread.php?t=50060
