Berakhot 45a:1-3

מַתְנִי׳ שְׁלֹשָׁה שֶׁאָכְלוּ כְּאַחַת, חַיָּיבִין לְזַמֵּן. אָכַל דְּמַאי, וּמַעֲשֵׂר רִאשׁוֹן שֶׁנִּטְּלָה תְּרוּמָתוֹ, מַעֲשֵׂר שֵׁנִי וְהֶקְדֵּשׁ שֶׁנִּפְדּוּ, וְהַשַּׁמָּשׁ שֶׁאָכַל כְּזַיִת, וְהַכּוּתִי — מְזַמְּנִין עָלָיו. MISHNA: This mishna sets out the essential halakhot pertaining to the invitation to recite Grace after Meals after a joint meal [zimmun]: Three people who ate as one are required to form a zimmun and recite Grace after Meals. If, among the diners, one ate doubtfully tithed produce [demai], and first tithe from which its teruma was already taken, or second tithe, and consecrated food that were redeemed and therefore permitted to be eaten; and even the waiter who served the meal to the diners and who ate at least an olive-bulk from the meal, and the Samaritan [Kuti] who ate with two others at a meal; each of these people is included among the three to obligate those with whom they ate in a zimmun.

אָכַל טֶבֶל, וּמַעֲשֵׂר רִאשׁוֹן שֶׁלֹּא נִטְּלָה תְּרוּמָתוֹ, וּמַעֲשֵׂר שֵׁנִי וְהֶקְדֵּשׁ שֶׁלֹּא נִפְדּוּ, וְהַשַּׁמָּשׁ שֶׁאָכַל פָּחוֹת מִכַּזַּיִת, וְהַנׇּכְרִי — אֵין מְזַמְּנִין עָלָיו. נָשִׁים וַעֲבָדִים וּקְטַנִּים — אֵין מְזַמְּנִין עֲלֵיהֶן. עַד כַּמָּה מְזַמְּנִין? — עַד כְּזַיִת. רַבִּי יְהוּדָה אוֹמֵר: עַד כְּבֵיצָה. However, one who ate untithed produce [tevel], and first tithe from which its teruma was not separated, and second tithe, and consecrated food that were not redeemed, and the waiter who did not eat an olive-bulk, and the gentile who ate with two Jews, none of these people is included among the three to obligate those with whom they ate in a zimmun. Women, slaves, and minors do not obligate those with whom they ate in a zimmun. How much must one eat to obligate those with whom he ate in a zimmun? An olive-bulk of food suffices to obligate those with whom they ate in a zimmun. Rabbi Yehuda says: An egg-bulk is the minimum measure to obligate those with whom they ate in a zimmun.

גְּמָ׳ מְנָא הָנֵי מִילֵּי? אָמַר רַב אַסִּי: דְּאָמַר קְרָא ״גַּדְּלוּ לַה׳ אִתִּי וּנְרוֹמְמָה שְׁמוֹ יַחְדָּו״. רַבִּי אֲבָהוּ אָמַר: מֵהָכָא: ״כִּי שֵׁם ה׳ אֶקְרָא הָבוּ גֹדֶל לֵאלֹהֵינוּ״. GEMARA: With regard to the basic mitzva of zimmun, the Gemara asks: From where are these matters derived, that after a meal in which three diners participated, a zimmun must be recited? Rav Asi said: As the verse states: “Praise God with me, and we will exalt His name together” (Psalms 34:4), i.e., the one reciting the blessing turns to at least two others to praise God together. Rabbi Abbahu said: The source of the mitzva of zimmun is derived from the verse here: “When I call the Name of the Lord, give [plural] praise to our God” (Deuteronomy 32:3).

אָמַר רַב חָנָן בַּר אַבָּא: מִנַּיִן לָעוֹנֶה ״אָמֵן״ שֶׁלֹּא יַגְבִּיהַּ קוֹלוֹ יוֹתֵר מִן הַמְבָרֵךְ — שֶׁנֶּאֱמַר: ״גַּדְּלוּ לַה׳ אִתִּי וּנְרוֹמְמָה שְׁמוֹ יַחְדָּו״. Having mentioned these verses, the Gemara cites related matters. Rav Ḥanan bar Abba said: From where is it derived that one who answers amen should not raise his voice louder than the one reciting the blessing? As it is stated: “Praise God with me, and we will exalt His Name together”; together and not with the respondent raising his voice louder than the one reciting the blessing.

אָמַר רַבִּי שִׁמְעוֹן בֶּן פַּזִּי: מִנַּיִן שֶׁאֵין הַמְתַרְגֵּם רַשַּׁאי לְהַגְבִּיהַּ קוֹלוֹ יוֹתֵר מִן הַקּוֹרֵא, שֶׁנֶּאֱמַר: ״מֹשֶׁה יְדַבֵּר וְהָאֱלֹהִים יַעֲנֶנּוּ בְקוֹל״, שֶׁאֵין תַּלְמוּד לוֹמַר ״בְקוֹל״, וּמָה תַּלְמוּד לוֹמַר ״בְקוֹל״ — בְּקוֹלוֹ שֶׁל מֹשֶׁה. Similarly, Rabbi Shimon ben Pazi said: From where is it derived that the translator who translated the public Torah reading into Aramaic is not permitted to raise his voice louder than the reader? As it is stated: “Moses spoke, and God responded in a voice” (Exodus 19:19). This verse requires further consideration, as there is no need for the verse to state: In a voice. The phrase, in a voice, adds nothing. Rather, to what purpose did the verse state: In a voice? In Moses’ voice, i.e., in a voice no louder than Moses’ voice. This verse instructs subsequent generations that Torah readers and translators should keep their voices at an equal volume just as Moses transmitted God’s word to the people and their voices were equal in volume.

תַּנְיָא נָמֵי הָכִי: אֵין הַמְתַרְגֵּם רַשַּׁאי לְהַגְבִּיהַּ קוֹלוֹ יוֹתֵר מִן הַקּוֹרֵא. וְאִם אִי אֶפְשָׁר לַמְתַרְגֵּם לְהַגְבִּיהַּ קוֹלוֹ כְּנֶגֶד הַקּוֹרֵא — יְמַעֵךְ הַקּוֹרֵא קוֹלוֹ וְיִקְרָא. This was also taught in a baraita: The translator is not permitted to raise his voice louder than the reader. The converse is also true; and if the translator cannot raise his voice to match that of the reader, the reader should lower his voice and read.

אִתְּמַר: שְׁנַיִם שֶׁאָכְלוּ כְּאַחַת, פְּלִיגִי רַב וְרַבִּי יוֹחָנָן. חַד אָמַר: אִם רָצוּ לְזַמֵּן — מְזַמְּנִין, וְחַד אָמַר: אִם רָצוּ לְזַמֵּן — אֵין מְזַמְּנִין. The mishna rules that three who ate as one are required to join together and recite Grace after Meals. The Gemara discusses this halakha further: It was stated: Two who ate as one and wish to join together in a zimmun, although they are under no obligation, are they permitted to do so? Rav and Rabbi Yoḥanan disagreed: One said: If they wanted to join together, they may form a zimmun. The other said: Even if they wanted to join together, they may not form a zimmun.

תְּנַן: שְׁלֹשָׁה שֶׁאָכְלוּ כְּאַחַת חַיָּיבִין לְזַמֵּן. שְׁלֹשָׁה — אִין, שְׁנַיִם — לָא! The Gemara cites a proof from what we learned in our mishna: Three who ate as one are required to join together and recite Grace after Meals. By inference: Three, yes, they form a zimmun; two, no, they do not form a zimmun. This contradicts the opinion that holds that two individuals who wish to form a zimmun may do so.

הָתָם חוֹבָה, הָכָא רְשׁוּת. The Gemara answers: There is no proof from the mishna, as there, the mishna discussed an obligatory zimmun; here, the amora’im disagree with regard to an optional zimmun.

תָּא שְׁמַע: שְׁלֹשָׁה שֶׁאָכְלוּ כְּאַחַת — חַיָּיבִין לְזַמֵּן, וְאֵין רַשָּׁאִין לֵיחָלֵק. שְׁלֹשָׁה — אִין, שְׁנַיִם — לָא! The Gemara cites an additional proof. Come and hear: Three who ate as one are required to join together and recite Grace after Meals and may not disperse to recite Grace after Meals individually. Apparently, three, yes, they form a zimmun; two, no, they do not form a zimmun. If a zimmun was possible with two people, three people would not be forbidden to disperse, as even if one recited Grace after Meals alone, the remaining two would constitute a zimmun.

שָׁאנֵי הָתָם דְּקָבְעוּ לְהוּ בְּחוֹבָה מֵעִיקָּרָא. The Gemara rejects this proof: It is different there, in the case of a group of three who dispersed, because from the outset, they established themselves as a group of three who were obligated to form a zimmun. Consequently, they are not permitted to forego an obligatory zimmun in favor of an optional one.

תָּא שְׁמַע: הַשַּׁמָּשׁ שֶׁהָיָה מְשַׁמֵּשׁ עַל הַשְּׁנַיִם — הֲרֵי זֶה אוֹכֵל עִמָּהֶם, אַף עַל פִּי שֶׁלֹּא נָתְנוּ לוֹ רְשׁוּת. הָיָה מְשַׁמֵּשׁ עַל הַשְּׁלֹשָׁה — הֲרֵי זֶה אֵינוֹ אוֹכֵל עִמָּהֶם, אֶלָּא אִם כֵּן נָתְנוּ לוֹ רְשׁוּת! The Gemara cites an additional proof. Come and hear, based on what was taught in a baraita: A waiter who was serving two people eats with them, although they did not give him permission to do so, because he will thereby be eligible to join them in a zimmun. If a waiter was serving three people, he may not eat with them unless they gave him permission to do so. Evidently, two may not form a zimmun. If that were the case, the waiter would require permission even when serving two people.

שָׁאנֵי הָתָם The Gemara responds: It is different there,