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Was the Torah given on Shavuos?
No! The first Shavuos Holiday was in fact celebrated the day before Matan Torah, as the Torah was given on Shabbos, the 51st day of the Omer, while Shavuos that year fell on Friday, the 50th day of the Omer. Nonetheless, the Holiday of Shavuos has become synonymous with the commemoration of Matan Torah in culture as well as in various matters of Halacha and Minhag, such as the prohibition to fast according to all opinions, reciting Zeman Matan Toraseinu in prayer, reading the section of the Aseres Hadibros in Kerias Hatorah, beautifying one’s home with flours and plants, staying awake throughout the night learning, eating dairy. The reason for this is because the Torah was given on the 6th day of Sivan, and in today’s calendar the 50th day of Sefira always coincides with the 6th of Sivan. Furthermore, there is also an internal connection between Shavuos, which is the 50th day of the Omer, and Matan Torah, as the Torah is fit to be received on the 50th day of preparation of Sefira. [See Shaar Halacha Chapter 1 for the full details on this matter.]
 Admur 468:22 “As on the day of Shavuos, before Matan Torah…” [See Shabbos 129b; M”A 468:15]; Admur 494:1 that in the year of the exodus, Matan Torah was on the 51st day of count, which would make Shavuos, the 50th day, fall on Erev Matan Torah.”; M”A 494:1; Chok Yaakov 494:1; See Aruch Hashulchan 494:1
 Admur 494:1; Rivash 96; Peri Chadash 494:1; Chok Yaakov 494:1; See Likkutei Sichos 3:997
 See Likkutei Sichos 3:997; The Divrei Nechemia Hashlamos 581 Kuntrus Acharon suggests that the reason we say Zeman Matan Toraseinu in Davening, and connect Shavuos with Matan Torah is because in truth the Torah was meant to be given on the 50th day of the Omer, which is the Shaar Hanun of Bina, and it is due to alternative factors that in the first year the giving of the Torah was delayed until the next day. Accordingly, each year, on the 50th day of the Omer, the revelations of Matan Torah take effect irrelevant to the day of the month, and therefore even if Shavuos falls on the 5th or 7th of Sivan one is to say Zeman Matan Toraseinu. The Rebbe ibid negates his opinion, although explains that there are two aspects of Matan Torah, one personal and the second global. The personal Matan Torah is on one’s 50th day of count, irrelevant of the date of the month. The global Matan Torah however only takes place on the 6th of Sivan each year, irrelevant to one’s day of count. The personal Matan Torah represents that time that one is fit to receive the Torah while the global Matan Torah represents when Hashem actually gives the Torah.