Residue Number System
Yet another number system, this one is quite funky. In Residue number systems numbers are represented by the string of the number modulo various primes. For example 4 in (2,3) residue number system is written as 012,3 and 510 = 122,3. The largest possible number that one can represent in residue number system defined by some numbers the is least common multiple of those numbers. When they are primes, it's just their product (since lcm of primes is always their product).
Clock
As with all number system, the best demonstraction is to show digital (or analog) clocks. I will slightly inefficient residue number system - for hours 6 and 8, for minutes and seconds 10 and 12. They ware chosen for one simple reason - their lcm matches the maximum value of slot that they fill: lcm(6,8) = 24, lcm(10,12) = 60.
Since we need 12 digits, I will use the convention of dozenal.