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... 

What you still want to show is that $lcm(lcm(a,b),c)|lcm(a,b,c)$. So what you want to show is that the least, or actually any, common multiple of $a$, $b$, and $c$ is a multiple of the least common multiple of $lcm(a,b)$ and $c$. This follows from the fact that a common multiple of two numbers is a multiple of their least common multiple.
