Some facts about fast days’ connection to seven
The numbers seven and three occur well over a hundred times in Judaism, as do their combination the number ten.
The Number Seven Symbolized Completeness to Many Jews
Ibn Ezra interpreted the oft-occurring number seven as a symbol of completeness. Three, on the other hand, is “always bad because it is half” (or, more precisely, close to half) of seven. It is the turning point to the change that will reach its climax with seven. Thus, in Genesis 34:25, the people of Shechem were sickest on the third day after their circumcision. The idea of seven signifying completeness is also the view of other writers, such as S. R. Hirsch.
Baruch Epstein agrees as well. He states that everything becomes complete with seven. Thus, a person naturally only feels full joy after a seven-day celebration, as in the seven days of a wedding celebration and the seven days of Passover and Sukkot. Similarly, he felt that mourning can only be complete and cathartic if a person has seven days of strict mourning, called shiva.
Thus, it is possible that the word sheva, “seven,” is related to sova. The latter means satisfaction and completeness, as in eating until a person is satisfied. An example is I Chronicles 23:1, which states that David “was old and sova in days.”
The fast of Tisha b’Av commemorates the destruction of the First and Second Temples. The First Temple was destroyed by the Babylonians on the seventh of the month of Av according to II Kings 25:8–9 and the tenth of Av according to Jeremiah 3:12, in 586 B.C.E. The Tosephta Ta’anit 4:10 and the Babylonian Talmud, Ta’anit 29a, explain the discrepancy: the outer walls were demolished on the seventh and the Temple itself on the tenth. In any event, we again see the numbers seven and ten.
The Second Temple
The Second Temple was destroyed by the Romans on the tenth of Av in 70 C.E., according to the historian and Jewish general Josephus. Josephus was present at the occurrence. But the Babylonian Talmud, Ta’anit 29a, several centuries later, gives the date as the ninth of Av. Thus according to Josephus at least, we have another seven (three plus seven).
The fast of Av
Why was a fast established on the ninth for both the destruction of the First and Second Temples? It is possible that the rabbis did not want to use seven because this number is associated with completeness, with creation, the Sabbath, and with joy, and the rabbis did not want to use seven and give the impression that Judaism was completely destroyed and that there was joy connected to the event. Eight may have not been chosen since, as we will show shortly, eight was seen as a symbol of a new beginning and the Temple destruction was the end of an era. Similarly, ten may have seemed inappropriate because, again as we indicated earlier, it is made up of seven and three, the two numbers connected with completeness.
It is also possible that the rabbis did not want to have the fast of Av on the tenth because the tenth would remind the Jew of the other fast of the tenth, Yom Kippur, and they did not want the Jews to compare both days. The fast of Av is considered an enormous and total calamity. Yom Kippur, on the other hand, has positive elements: during the Temple days, the afternoon of Yom Hakippurim, the biblical name, was a happy time when young boys and girls would meet and dance. Additionally, on Yom Kippur one is able to seek forgiveness for past misdeeds.
The remaining three smaller fasts are also associated with the number seven or its derivatives three and ten. The Fast of Gedalia occurs on the third of Tishrei. The fast of Asarah b’Tevet is on the tenth. The fast of Shiva Asar b’Tammuz is on the seventeenth, which is seven and ten.
If seven is a symbol of completeness, then eight denotes the beginning of something new. A male child is circumcised after the completion of seven days of life, on the eighth day. Similarly, the holiday of Shemini Atzeret occurs after the completion of the seven days of Sukkot. The rabbis conceive it as a renewal of a relationship with God.
 See Israel Drazin, Maimonides and the Biblical Prophets, Gefen Publishing House, 2009, “The Significance of the Number Seven,” pages 277-303.
 Commentaries to Exodus 3:15; Leviticus 22:27, 23:24, 26:18; Numbers 23:1; and Deuteronomy 28:7.
 Commentary to Leviticus 12:2, 3.
 Tosaphot Berakhah, book one, pages 87–89, and Mekor Barukh, part 3, chapter 26.
 Wars 6:249–250.
 Yom Hakippurim was a day when the high priest would bring offerings and atone for his misdeeds, those of his family, and the nation. When the temple was destroyed and sacrifices ceased, Yom Hakkipurim could no longer be observed. The rabbis invented Yom Kippur, a day when individuals, not the High Priest, would fast and consider their own misdeeds and resolve to improve. See Israel Drazin, Mysteries of Judaism, Gefen Publishing House, 2014, “Yom Kippur: A Holiday the Romans Destroyed,” pages 16-18.
 S. R. Hirsch, ibid, note 3.