Of course you will not proceed here unless you have managed to get through all my pages describing the week. You start with the week (from a to c) and then you will be prepared to go on with the month in a similar way.
The Sumerians based their system of counting on 60 and in the extraordinary comprehensive book by Ifrah he explains their choice of this high number in about a dozen different ways, none of them convincing as such. Taken all together, though, they may seem convincing.
But nowhere in Ifrah's book have I found the most reasonable explanation:
In the beginning the methods of counting did not manage more than whole positive numbers. Therefore it was very inconvenient that the period of the phases of the Moon took more than 29 days but less than 30. Keeping order in the calendar for the Moon required some kind of trick.
The trick established all over the ancient world (as I guess) was to count two periods, because with 59 days the problem was solved. The revolution of the phases of the Moon is very nearly 29 ½ days.
And day no. 60 then became the first day of the new double-moon period. The Sumerians therefore had the same sign for 60 as for 1. Zero was not yet invented and there could be no misunderstanding if you knew what you were talking about (as people did at that time).