A divisibility test for 37 and 111

Starting from the last three digits of the number, separate it into groups of three digits in the same way you did for two digits. Add the groups of three digits or less, and feel free to repeat the process for the result if it's too unwieldy. If the result is divisible by 111 or 37 then the initial number is divisible by 111 or 37.

Example: Take 1369 (37^2): 1 369, 369 + 1 = 370, obviously divisible by 37.

Another example: Take 1367643 (111^3 + 12): 1 367 643, 643 + 367 = 1010, 1010 + 1 = 1011, 1 + 11 = 12. This shows that the remainder of the division can also be found.