The Jews responded to this news with a total of six actions: they mourned, fasted, cried, eulogized, and donned sack and ash. The M’nos HaLevi writes that there is significance to this number. These six actions correspond to the six days in which the Jews participated in Achashverosh’s party (see Esther 1:5). Indeed it was a seven-day party, and the Jews took a break from the last day because it was Shabbos.
Since the verse that describes Achashverosh’s party (1:4), the verse says the party lasted for many days (yamim rabim), and gives the number of days as 180, R’ Yehonason Eibshutz wonders why the phrase “yamim rabim” is not superfluous. He answers that this phrase refers to the kind of days they were, long summer days, concluding with Yom Kippur. This is the day on which no Jew sins. In fact, he adds that the gematria of the Satan (hasatan) is 364 (5+300+9+50), one less than the total amount of days in a solar year, indicating that the Evil Inclination has no hold on us for one days out of the year – Yom Kippur. Therefore, there were only six days for which the Jews needed to atone.
The Ginzei HaMelech writes that the Sfas Emes views Megillas Esther as the beginning of the Oral Law. Mordechai was even a member of the Anshei Kenesset HaGedolah (Men of the Great Assembly) that began the establishment of Rabbinic law. The Oral Law is represented by the number six, as that is the total number of Orders in the Mishnah – Seeds, Festivals, Women, Damages, Holy Items, Purity. The Jews mourned in six different ways in to show their new-found reverence for the Oral Law.
Interestingly, according to the Vilna Gaon, there are not six actions here, but five. In his understanding, the great mourning is not a separate action, but is one general action described with the remaining five detailed descriptions. According to him, these five correspond to the five actions Jews are supposed to take (Mishnah, Taanis 1:3-7) when they are suffering agriculturally.