Is $\mbox{lcm}(a,b,c)=\mbox{lcm}(\mbox{lcm}(a,b),c)$? $\newcommand{\lcm}{\operatorname{lcm}}$Is $\lcm(a,b,c)=\lcm(\lcm(a,b),c)$?
I managed to show thus far, that $a,b,c\mid\lcm(\lcm(a,b),c)$, yet I'm unable to prove, that $\lcm(\lcm(a,b),c)$ is the lowest such number... 

The "universal property" of the $\newcommand{\lcm}{\operatorname{lcm}}\lcm$ is

if $\lcm(a,b) \mid x$, and $c \mid x$, then $\lcm(\lcm(a,b),c) \mid x$.

Make a good choice for $x$, for which you can prove the hypothesis and/or the conclusion is useful.
