Here is a test to determine whether a number is divisible by 19:
Remove the last digit of the number, add twice that digit to the truncated number, and keep repeating this procedure until a number less than 20 is obtained. The original number is divisible by 19 if and only if the final number is 19. For example, the calculation below and to the left shows that 25549 is not divisible by 19, while the one on the right shows that 47063 is divisible by 19.
Source: USA Mathematical Talent Search