When will the stars be right again? (Messing around with a fictional calendar)

[Yora]

I am working on a calendar for a fantasy world, and it's pretty simple:
The new year begins at dawn on the morning after the first moon after midwinter night. No weekdays, no standard month length to worry about.

Now I made up some numbers for the orbits of the planet around the sun and for the moon around the planet, just by picking numbers that "look nice".
A solar year is very close to 372.35 days, which means that there will be a leap year every three years. Except that there is only a single regular year between the 6th and 7th leap year of every circle. That seems to be entirely the result of of the .35 and I like this particular effect on the calendar.
I also picked the duration for a lunar month to be close to 29.2 days. That means you get 4 months of 29 days and every 5th month with 30 days. Also neat.

Now here's the million dollar question: How often will the midwinter night be a night of a new moon?

Or to put it more mathmatical, how often will a 372,35 cycle and a 29,2 cycle match up exactly? I think it has to be a pretty basic calculation, but I'm not quite sure how to do it. I tried 372.35 times 29.2 and multiplied that by 100 to get rid of the decimal points, which results in 2,920 years.
But as further complication, I think that's the the number of nights between "perfect alignments at midnight". But the calendar is not calibrated to midnight, but to the dawn after midwinter night. If the "perfect new moon alignment" happens at 4:48 PM or 7:12 AM, that still counts.
So, do I just have to divide the 2.920 years by 3?
That gets me 973.3333 years. Which in turn would mean two cyles of 973 years followed by one cycle of 974 years.

Is this math correct?

[Max™]

Not sure, BUT, it occurs to me that you might be able to find the answer or at least have fun playing with this: http://www.edtechbybowman.net/PhysAstroSims/orbits/PlanetOrbit.html and inputting your chosen values.

Whoops, grabbed the wrong tab, meant this one: http://dan-ball.jp/en/javagame/planet/ though the other one can still be interesting.

[NichG]

So the two cycles correspond to the integers 37235 and 292 in terms of hundredths of a day. Lets factor those into primes:

37235: 5 * 11 * 677
292: 2 * 2 * 73

So since they don't have common factors to divide out, they line up every 292 * 37235 hundreds of a day, as you surmised.

But lets say you rounded to tenths of a day instead, and just wanted to know how often they line up to that degree of accuracy. Then the numbers are:

3724: 2*2*7*7*19
29: 29

So again, no common factors. So they line up every 29*3724/10 days, which is about 10 times as often as they line up to a hundredth of a day (makes sense, since the range of times that correspond to a 'match' is ten times larger, so random numbers fall in that range ten times as often).

Three hours is about a tenth of a day, so you'd expect them to line up to a 3 hour range about once every hundred years.