Shrajan's geometric progression

The numbers are in Geometric Progression. This indicates that they are of the form- $a, ar , ar^{2}$ . Their Product is $a^{3}r^{3} = 216$ . Thus we get, $ar$ =6. Rewriting the solution for their sum, that is- $a+ar+ar^{2}=19$ in terms of $ar$ , we get that the terms are 4,6,9. Their LCM is $36$

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Since the numbers are in geometric progression, we can express them as: $\frac { a }{ d } ,a,ad$

Multiplying them gives us ${ a }^{ 3 }$ which is $216$ ======= $a=6$

Now we have $\frac { 6 }{ d } ,6,6d$

Adding them gives us $\frac { 6 }{ d } +6+6d=19$

This simplifies to ${ 6d }^{ 2 }-13d+6=0$ ========= $x=1.5$

Substitution gives us the three numbers: $4,6,9$ with a common ratio of $1.5$

the $lcm$ of these numbers is $\boxed{36}$