The Shulchan Aruch (Orach Chaim 124:4) writes that if 9 men don’t answer “omein” to a berachah during chazoras hashatz it is close to a berachah l’vatalah. There is also a halachah that if men leave the shul during the chazoras hashatz, and only 6 men remain, the shaliach tzibur can continue until (at least) the end of the amidah (Shulchan Aruch, Orach Chaim 55:2), and this applies even if they leave during the course of the first berachah (Mishnah Berurah 55:8). In such a case it will be impossible for 9 men to answer “omein” to even one of the berachos.

Although the Bavli only gives the 10 evil meraglim as the source of the number 10 for a minyan, the Yerushalmi (Megillah, perek 4, halachah 4) gives a second source, again using the word “betoch” (Bereshis 42:5). Here the number 10 is derived from the sons of Ya’akov, (minus Yosef and Binyamin), who went down to Egypt to purchase food.