A minyan, the presence of ten men, is required for those parts of the prayer service that are deemed Devarim She-bikdushah (literally, words of sanctifying). These include Kaddish, Barchu, Kedushah, the repetition of the Amidah, and (according to some) the reading of the Torah, the reading of the Haftarah, and the priestly blessing. This rule is derived from the verse, “I will be sanctified among the children of Israel” (“Ve-nikdashti be-toch Bnei Yisrael”) (Vayikra 22:32). How do we know that the number referred to here is specifically ten, neither more nor less?
One way of arriving at ten is through a gezeirah shavah. (This is a method of rabbinic exegesis in which a similar word appearing in two different contexts is used to infer that the details of one context apply to the other.) The word “toch” (“among”) appears in the verse about sanctifying G-d, and in the story of Korach’s rebellion. Regarding the latter, the Torah states (Bamidbar 16:21), “Separate yourselves from among (mi-tokh) this community (edah).” However, there the Torah is referring to a group of 250 people. How is it useful for arriving at the number ten?
This involves a bit more exegesis. The word “edah,” which is used in the story of Korach, is also used in reference to the ten spies who spoke badly of the Land of Israel, as we read (Bamidbar 14:27), “How much longer will that wicked community (edah) keep muttering against Me?” We see that the definition of a community is ten. Thus, the community within which we sanctify G-d’s name must be similar to the spies (not in their sinfulness, of course, but in being free, adult males).
It should be noted that the above is not a combination of one gezeirah shavah with another (toch-toch and edah-edah), which would possibly break a rule of exegesis. Rather, we learn from the case of the spies in Parshat Shelach that the definition of the word “edah” is ten everywhere it appears. This includes the verse in Parshat Korach, where the word “toch” is associated with ten (through the word “edah”). And a gezeirah shavah (toch-toch) connects that verse with the verse about sanctifying G-d.