\(\bullet\) LCM of any two non-zero rational numbers always exists.
\(\bullet\) LCM of any non-zero rational and any irrational number never exists.
\(\bullet\) LCM of any two irrational numbers may or may not exist.
\(\bullet\) Also you can start out the discussion here on this note through comments.
Till then you can try to solve the set of such problems : IIT Foundation Classes