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a copy of a few lines from the halacha:
Similarly, when the days of a lunar year are counted in groups of seven, according to the weekly cycle, there is a remainder of four days, eight hours, and 876 units (in numerical terms, 4 - 8 - 876).
question how do you calculate 4-8-876 for a year?
The Rambam is trying to explain how the end of the year "drifts" across the days of the week.
The lunar year (twelve lunar months) is roughly 354.3 days. 350 is divisible by seven (hence "groups of seven"), so the only thing that decides what day of the week the year ends on is the remainder. --- the extra 4.3 days.
Thus, if the end of the previous year was on a Sunday, the end of the current year will be on Thursday (~4.3 days later).
Every halachic hour is divided into 1080 chalakim or parts.
e.g. a month is 29 days 12 hours and 793 chalakim. ( I knew that and looked up about the chalakim to remind myself/check about the 1080)
so you could work out 876/1080 for what faction of an hour, then times by 60 to get minutes
52.56 minutes. that's 52 minutes and 33.6 seconds.
so 876 chalakim are 52 minutes 33 seconds, if my maths is right.
I can't comment on the rest of what you wrote re groups of seven.. off the top of my head not sure what that is about(4 days etc). I remember about the difference between the lunar and solar calendar being about 11 days. The lunar year finishing 11 days soon. So an extra lunar month at the end of every third lunar year pushed it out to sync it, the famous 7 lunar months in 19 years(is approx 1 extra lunar month every 3 lunar years).
If you're seeking "programmatics", i.e. how you would program these calculations using a computer as well as the math involved, you may want to read this page as well as some of the links within it.
If this is not it, please clarify what you are seeking, and I'll try to answer or redirect you, correctly.
The routine is:
[Month remainder] * 12 - [full weeks] = result
[1 - 12 - 793] * 12 => 18 - 8 - 876 - 14(2 weeks) = 4 - 8 - 876
1.53059413580247(Month remainder) * 12 => 18.3671296296296 - 14(2 weeks) = 4.3671296296296 = 4 - 8 - 876
