Why Are Ten Men Required for a Minyan?

The requirement is derived from the verse "And I shall be sanctified in the midst of the children of Israel" — all matters of sanctity, such as Kaddish, Birkhu, and similar prayers, must be performed "in the midst of the children of Israel." How many constitute "the children of Israel"? We learn a gezerah shavah (equivalent phrase) between the two occurrences of "in the midst" from the verse "Separate yourselves from the midst of this congregation" where the spies were ten in number (excluding Joshua and Caleb). Ten is the highest complete number and symbolizes a complete structure. (Numbers also contain a hidden code, as with the Hebrew letters, and each number has its internal significance.) Ten Jews represent the completeness of the structure of the nation of Israel, which causes the Divine Presence and sanctity to rest upon those ten, enabling them to sanctify the great and revered Name at that moment.