Rambam's method for calculating the conjunction of the moon (and sun)

Question

I think I'm missing something basic here...

The Rambam, in Hilchot Kiddush Hachodesh (6:3) says that the interval between one conjunction to the next is is 29 full days, 12 hours of the 30th day, and 793 units of the 13th hour. This, he says, is the length of the lunar month.

But later, after making the point in 6:5 that dividing a lunar month into groups of seven (i.e. weeks) yields a remainder of 1 day, 12 hours, and 793 units, he then goes on to say (6:6) that

When you know the time of the conjunction [of the sun and the moon] for any particular month, and add 1 - 12 - 793, you will arrive at the time of the conjunction of the following month.

Again, I think I'm missing something basic here: why do you have to add that remainder to calculate the next molad? I don't understand, why, in the first place we're dividing a lunar month into groups of seven? Seems arbitrary to me. So, if the interval from one molad to the next is 29 full days, 12 hours of the 30th day, and 793 units of the 13th hour, then why does he later to say to add 1 - 12 - 793?

Answer 1

The Rambam here is giving an easy way to calculate the moment of conjunction for future months. A (lunar synodic) month, as you stated, is 29 days 12 hours and 793/1080 hours. Since 28 days is exactly 4 weeks (bringing us back to the same point in the week), the next conjunction will occur 1 day 12 hours and 793/1080 hours later in the week relative to the previous conjunction. This is useful because usually you can know approximately when the next conjunction will be. By adding 1-12-793 to a given conjunction time repeatedly, one can easily calculate a later conjunction time without constantly canceling out 7 days.

The Rambam (Kiddush HaChodesh 6:5) also gives similar mnemonics for figuring out the conjunction of a given month in later years. For calculating conjunction after 12 months (a regular year) add 4 days 8 hours and 876/1080 hours. For calculating conjunction after 13 months (a leap year) add 5 days 21 hours and 589/1080 hours. (And you can check by finding the difference between those two to be exactly 1 day 12 hours and 793/1080 hours which is from the addition of 1 extra month.) He also gives a mnemonic for the passage of 19 years (the Machzor -- cycle of leap years) to be 2 days 16 hours and 595/1080 hours.