For simplicity, let us work only with positive integers.

We want to prove that if a,b,m are integers, then $$ \lcm(ma,mb) = m\lcm(a,b). $$

First notice that any common multiple of $ma$ and $mb$ is also a multiple of $m$ so any common multiple of $ma$ and $mb$ is of the form $mk$ with some integer $k$

Now notice that if $t=\lcm(a,b)$ and $u<t$ it cannot happen that $a\mid u$ and $b\mid u$ since $t$ is the smallest number, So, when $a\nmid u$ then $ma\nmid mu$ and if $b\nmid u$ then $mb \nmid mu$ We conclude that $mu$ is not a common multiple of $ma$ and $mb$ when $u<t$

So far, we proved that $mt = m\lcm(a,b)$ is a common multiple of $ma$ and $mb$ and previous paragraph shows that there is no smaller common multiple, therefore $m\lcm(a,b)$ is the least common multiple of $ma$ and $mb$ in other words: $$ \lcm(ma,mb) = m\lcm(a,b). $$