jmc

Both $a$ and $b$ must be divisible by $6$, so the choices for $a$ are $$12,42,72,102,132,\dots$$and the choices for $b$ are $$24,54,84,114,144,\dots .$$We know that $\mathrm{lcm}[a,b]\cdot \gcd (a,b)=ab$ (since this identity holds for all positive integers $a$ and $b$). Therefore, $$\mathrm{lcm}[a,b]=\frac{ab}{6},$$so in order to minimize $\mathrm{lcm}[a,b]$, we should make $ab$ as small as possible. But we can't take $a=12$ and $b=24$, because then $\gcd (a,b)$ would be $12$, not $6$. The next best choice is either $a=12,b=54$ or $a=42,b=24$. Either of these pairs yields $\gcd (a,b)=6$ as desired, but the first choice, $a=12$ and $b=54$, yields a smaller product. Hence this is the optimal choice, and the smallest possible value for $\mathrm{lcm}[a,b]$ is $$\mathrm{lcm}[12,54]=\frac{12\cdot 54}{6}=2\cdot 54=\boxed{108}.$$