Explanation:

Let the numbers be x and (2000 - x).
Then, their L.C.M. = x (2000 - x).
So, x (2000 - x) = 21879
=> x² - 2000x + 21879 = 0.
=> (x - 1989) (x - 11) = 0,
=> x = 1989 or x = 11.
Hence, the numbers are 1989 and 11.

OR

Can easily be picked with the options.