Mishnayos Brachos Perek 7 Mishnah 4
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ברכות פרק ז׳ משנה ד׳
Three people who ate as one are not permitted to divide and recite Grace after Meals individually; rather, they recite the zimmun together. And the same is true of four who ate together, and the same is true of five. However, a group of six, up to but not including ten people who ate as one, may divide into two groups, each reciting its own zimmun. And a group of ten may not divide into two groups until there are twenty people present. The general principle is that a group may not divide unless the smaller groups will be able to recite the same zimmun formula that the whole group would have recited.
שְׁלשָׁה שֶׁאָכְלוּ כְאֶחָד, אֵינָן רַשָּׁאִין לֵחָלֵק, וְכֵן אַרְבָּעָה, וְכֵן חֲמִשָּׁה. שִׁשָּׁה נֶחֱלָקִין, עַד עֲשָׂרָה. וַעֲשָׂרָה אֵינָן נֶחֱלָקִין, עַד שֶׁיִּהְיוּ עֶשְׂרִים:
Bartenura
אינן רשאים ליחלק. מכיון שחלה עליהם חובת זימון:
וכן ארבעה וכן חמשה. אין השלשה מזמנין לעצמן והיחיד יחלק מהן, דאיהו נמי אקבע בחובת זמון:
ששה נחלקים. כדי זמון לכאן וכדי זמון לכאן, עד עשרה, אבל י׳ אין נחלקים דאתחייבו להו בזמון שיש בו הזכרת השם, עד שיהיו עשרים, ואז יחלקו לשתי חבורות אם ירצו:
אינן רשאין לחלק – Since an obligation of invitation [to a group recitation of ברכת המזון] falls upon them.
וכן ארבעה וכן חמשה – The “three” [of that group of four or five] (cannot separate) to recite ברכת המזון for themselves and the individual should separate himself off from them, for he also has an established obligation for the appeal to participate in the group recitation [of ברכת המזון], it says Psalms 68:27), “In assemblies bless God, [the LORD, O you are from the fountain of Israel].” And the determined Halakhic practice is that from three individuals until ten individuals, but not including ten, the person reciting the blessing says, “Let us thank Him whose food we have eaten/נברך שאכלנו משלו,” and everyone responds, “ברוך שאכלנו משלו ובטובו חיינו/Blessed is [He] whose food we have eaten and through whose goodness we live.” And with ten or more than ten people, the leader says, “נברך לאלהינו שאכלנו משלו/Blessed is our God whose food we have eaten.” And everyone answers: “ברוך לאלהינו שאכלנו משלו ובטובו חיינו/Blessed is our God whose food we have eaten and through whose Goodness we live.”
ששה נחלקין – in order to have a group recitation (of three individuals for ברכת המזון) and another equivalent group over there, until one reaches the number “ten.” But with ten individuals, we do not divide the group since it would obligate them to a have a group recitation including the mention of God’s name until there would be twenty and only then could they divide into two [separate] groups if they desired.