Another alternative for the remainder modulo 19

Another way of finding the remainder modulo 19 is as follows: Take the leftmost digit of the number, multiply it by 10 and subtract it from the next digit, and then add the leftmost digit to the result. Continue until the digits are exhausted. For example, consider the number 175. 7 - 10 = -3, -3 + 1 = -2. 5 - (-20) = 25, 25 +(-2) = 23 = 4 (mod19). That follows because 175=171 + 4 and 171=9 * 19. Another example: Let the number be 5777. 7 - 50 = - 43, - 43 + 5 = - 38. 7 + 380 = 387, 387 - 38 = 349. 7 - 3490 = - 3483, - 3483 + 349 = - 3134. - 1 - ( - 30 ) = 29, 29 - 3 = 26 = 7 ( mod 19 ). - 3 - 70 = - 73, - 73 + 7 = - 66 = 10 ( mod 19 ). - 4 - 100 = - 104, - 104 + 10 = - 94. - 94 + 5 * 19 = - 94 + 95 = 1 ( mod 19). So 5777 = 1 ( mod 19), which was to be expected,since both 57 and 76 are divisible by 19.