question #1 Sahih Muslim 1084b, Book 13, Hadith 28 : How does the Islamic calendar account for months with different numbers of days?

The Apostle of Allah (ﷺ) separated himself from his wives for a month. (His wives said: ) He came to us on the morning of the twenty-ninth. Upon this some, of the people said: It is the morning of twenty- ninth (according to our calculation). Upon this the Messenger of Allah (ﷺ) said: The month. may also consist of twenty-nine days. The Apostle of Allah (ﷺ) then flapped his bands thrice, twice with all the fingers of both his hand (to indicate twenty-nine) and by the third time with nine (fingers).