I want to do arithmetic in base 60, but since remembering 60 different symbols is difficult, I want to represent each digit by its value mod 3, 4, and 5 (equivalent by the Chinese remainder theorem). Adding the digits is as simple as adding each modulus separately. However, knowing when to carry one to the next higher power of 60 is something I haven't figured out yet.
The question is this: If I know the remainders mod 3,4,5 of two nonnegative integers x,y < 60, is there an algorithm for determining if x+y >= 60 without computing the values of x,y?