Set A consists of integers {3, -8, Y, 19, -6} and Set B

I think the question should read: "number M represents the median of set B"

Set A: as y>21 then if we arrange numbers in ascending order we'll get {-8, -6, 3, 19, y}. Now, the median of a set with odd # of terms is just a middle term (when ordered in ascending /descending order), so L=median=3;

Set B: {-5, -3, 0, k, 9, 16}. The median of a set with even # of terms is the average of two middle terms, so depending on k, M=median can take different values;

Next, we are told that L^M=3^M is a divisor of 26. As 26 is not a power of 3 then either M=0, and in this case 3^M=3^0=1 and 1 is a divisor of every integer OR M is some irrational number. But as k is an integer, then median of set B can not be an irrational number, so it must be 0. In order median of {-5, -3, 0, 9, 16, k} to be zero k must be zero too: {-5, -3, 0, 0, 9, 16} --> median=M=(0+0)/2=0.

Answer: C.

If it were as you wrote - "number M represents the mode of set B", we would have the following:

The mode is the number that occurs the most frequently in a data set and set B is {-5, -3, 0, 9, 16, k} so k must equal to one of the other numbers in a set in order it to have a mode, from answer choices k can only be 0 (as no other number from set B is represented there).

Answer would still be C, though in this case all other information in the stem would be redundant.

Hope it's clear.