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Why a Minyan?

What is a Minyan and Why is it Important?

"Traditionally, it is necessary for ten males above Bar Mitzvah age to be present to hold a religious service or to proclaim the Holiness of G-d and His kingship at public occasions, such as weddings, when marriage blessings are pronounced. Ten men constitute a quorum (a minyan in Hebrew).

Many theories have been advanced for the selection of this number.The Rabbis of the Talmud (Megila 23b and Berachot 21b) explain that because the ten (evil) spies in Numbers 14:26 are referred to as an "edah", a "congregation," ten is the number that constitutes a minyan." [From The Jewish Book of Why, Kolatch, p. 141].

Genesis chapter 18 tells the story of the Angels letting known Abraham that Sodom and Gomorrah (and three other cities: Admah, Zeboiim and Zoar) would be destroyed, and how Abraham bargained with them in case there were righteous men in those cities. Abraham questioned them: "Wilt Thou indeed sweep away the righteous with the wicked?" And the number of possible righteous went from fifty to ten. In the Artscroll Chumash (page 83, comentary 24) explains that Abraham asked first for fifty, making ten men per city, thinking that ten righteous men would be enough to save that city. But Abraham's plea was that the succesively smaller number of righteous men should be sufficient to save them all.

When a Minyan is present, the "Barechu" is recited, along with Kaddish
and after the Amidah prayer is done silently and individually, the cantor
does the public repetition. 
The Torah reading only occurs when there
is a Minyan present.