The Mayan calendar system uses a number of different periodic processes, and provides a simple but very important example of a practical use of the CRT.

The Tzolkin, or Day Count, has twenty weekdays (Ik, Akbal,... Auau) and thirteen numbers, 1-13. Each day, the day name advances, and so does the number. For example, 7 Ik is followed by 8 Akbal. These name/number pairs repeat in a 260 day cycle, which has been in continuous uniterrupted use since at least 600BC.

The Haab, or Vague Year, is a 365 day year consisting of 19 months (Pop, Uo, ..., Cumku, Uayeb). The first 18 months have 20 days and Uayeb has five days. The Haab runs 0 Pop, 1 Pop, ..., 19 Pop, 0 Uo, 1 Uo,..., 4 Uayeb and then repeats to 0 Pop.

Together, the Tzolkin and Haab form the calendar round, with dates given by Tzolkin then Haab, for example, 7 Ik 0 Cumku. This cycle repeats every 18980 days, about 52 years, which means that a calendar round date is good for most practical purposes (such as birth dates).

The earlier Mayan period, from around the 1st century BC to the 13th century AD, also featured a system known as the long count, recently made somewhat famous by the fact that it finished a 5126 year cycle on December 21, 2012.
Long count dates have Kin (days) which run 0-19. 20 Kin make one Uinal, 18 Uinal make one Tun, 20 Tun make one Katun, and 20 Katun make one Baktun. Dates are written with Baktun first, as, for example, 9.7.17.12.14. After 13 Baktun, the date goes back to zero, so that 12.19.19.17.19 was followed by 0.0.0.0.0.

One major problem with studying the Mayan calendar is that the long count dates fell out of use hundreds of years before the Spanish arrived, and it is nontrivial to decide which Mayan long count dates correspond to which dates in the modern western calendar system - the 'correlation problem'.

The key document, the Chronicle of Oxcutzcab, says that a tun ended 13 Ahua 8 Xul in AD 1539, thus tying together the long count (tun ending), the calendar round, and the Julian calendar. From ancient records, the long count is known to have begun on 4 Ahua, 8 Cumku. So, given 0.0.0.0.0 = 4 Ahua 8 Cumku, one needs to solve $x$.0.0 = 13 Ahua 8 Xul. The day number gives the equation $360 x \equiv 9 \pmod{13}$. Since 8 Xul is 125 days after 8 Cumku, the Haab gives the second equation $360 x \equiv 125 \pmod{365}$.

So, there's a simple little use of CRT: Solve for $x$, and find $x \equiv 924 \pmod{949}$.

To finish the story, the year AD 1539 contains the long count tun 924 = 2.6.4.0.0 plus some multiple of 949 tun = 2.7.9.0.0. There is enough historical evidence to guess the date to within 949 tun (about 935 years), and so one learns that 11.16.0.0.0 is in AD 1539. Finally, the calendar round is still in use and so one can determine that 11.16.0.0.0 is November 12, 1539. I'll leave it as an exercise to determine that December 21, 2012 really was 0.0.0.0.0.