Prove that lcm$(a,b)$ = $k$·lcm$(a/k,b/k)$

Question

I have problem approaching this equation from definition directly. I know there's some kind of bridge connecting there three terms, but I just cannot see it. Thank you!

Answer 1

$[x,y]:={\rm lcm}(x,y).\,$ Using lcm universal property and basic divisibility properties, if $\,k\mid a,b\,$ then

$$[a/k,b/k]\mid n\iff a/k,b/k\mid n\iff a,b\mid kn\iff [a,b]\mid kn\iff [a,b]/k\mid n$$

Therfore $\ [a/k,b/k] = [a,b]/k\,$ since they divide each other by above.