Q. Why is Shavu’ot on 6 Sivan since no date is given in the Torah for the festival?
A. It was worked out on the basis of hints in the Torah. The Israelites came to Mount Sinai in the third month after they left Egypt. In that month is Shavu’ot. We have to count seven weeks of the Omer beginning on the second day of Pesach, and the seven weeks bring us to 6 Sivan.
A further significance applies to the sixth day, since the sixth day of Creation is when man was made, and only when man received the Torah on the sixth day of Sivan was the Creation able to be completed morally.
In addition, there were six hundred thousand Israelites (apart from women and children) at Mount Sinai, and the Torah is said to have six hundred thousand Hebrew letters.