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The takbīrs on the days of Tashreeq after the prayers are legislated and reported from the Sahābah (radiyallāhu ‘anhum). As for the report ascribed to the Prophet (salallāhu ‘alaihi wasallam), then it is weak.
So the takbīrs are recited in the ten days of Dhul-Hijjah beginning with the first day.
It is reported from Ibn ‘Umar and Abu Hurayrah (radiyallāhu ‘anhumā) that they would go out to the market in the ten days making the takbīrs and the people would make takbīrs due to their takbīrs. The Sahābah, ‘Umar (radiyallāhu ‘anhu) and a group of them would make takbīrs after the daily prayers beginning on the day of ‘Arafah until the last day of Tashreeq (13th Dhul-Hijjah).
‘Umar (radiyallāhu ‘anhu) would make takbīr in his tent in Minā and the people would hear that and also make takbīr with his takbīr. So the takbīr is legislated in the Sharī’ah but it is not obligatory. It is the Sunnah to do so on the day of ‘Arafah, the day of Nahr (‘Eid al-Adhā), and throughout the days of Tashreeq, at specific times after the prayers and at other times during the night and day.
As for the 1st till the 8th day of Dhul-Hijjah, then that is a time for general takbīrs without specifying them after the prayers. A person makes the takbīrs while in the street, in his home, in bed, etc. Then on the day of ‘Arafah and onwards, the Muslims make takbīrs in the streets, in the masjids, in the markets and at the end of each of the prayers. That which the majority of the scholars are upon is that this is a Sunnah (not obligatory).
The method of reciting the takbīr can be with an even number of takbīrs:
الله أكبر الله أكبر لا إله إلا الله، الله أكبر الله أكبر ولله الحمد
And some of the Sahābah would recite the takbīr with an odd number of takbīrs:
الله أكبر الله أكبر الله أكبر لا إله إلا الله، الله أكبر الله أكبر الله أكبر ولله الحمد
All of it is good whether it is even or odd. Also, one can recite:
الله أكبر كبيراً، والحمد لله كثيراً، وسبحان الله بكرة وأصيلاً
This is reported in narrations from the Sahābah and from the Prophet (salallāhu ‘alaihi wasallam). So, every person should recite by himself ― there is no reciting in a chorus (in jamā’ah), each person recites by himself, not in jamā’ah―and if it happens that a person’s voice coincides with the voice of his brother, there is no harm in that. As for a conducted recitation of the takbīrs starting together and ending together, then that has no basis in the religion.
See https://binbaz.org.sa/fatwas/8774 (Abridged)
Takbīr of Ibn Mas’ūd (may Allah be pleased with him)
الله أكبر، الله أكبر، الله أكبر، لا إله إلا الله، والله أكبر، الله أكبر، ولله الحمد
Ref: Musannaf Ibn Abi Shaybah: 4/195-196 no. 5679 (Abdullāh bin Mas’ūd (radiyallāhu ‘anhu) would make takbir from Fajr on the day of ‘Arafah till ‘Asr prayer on the Day of Nahr)
So throughout these days, Allah is plentifully remembered. It is reported by Ibn Abi Shaybah and Al-Hākim that, “Ali (radiyallāhu ‘anhu) would make the takbīrs after Fajr on the day of ‘Arafah until ‘Asr on the last day of Tashreeq and he would make it after ‘Asr also [on that day].” (See Al-Irwā, no. 653-654)
So, in conclusion, the takbīrs should not be restricted to just after the prayers, rather they should recited at all times and after the prayers.