A Digit Transformation

Extract

We describe a variant of the game with digits discussed in a previous article. There we considered a transformation of any number in a scale s + 1 in which each digit is replaced by a digit equal, modulo s, to the sum of the remaining digits. Starting for instance with the number 16427, in the scale of 10, the first digit remains 1, since 1 = 6 + 4 + 2 + 7 (mod 9), the second digit is replaced by 5, since 5 = 1 + 4 + 2 + 7 (mod 9), and so on, forming the number 15794; repeating the transformation we obtain in turn 73184, 72461, 49751, 48137, 16427, the original value recurring after six transformations. We showed that under this transformation any number x, with m digits in the scale s + 1 (none equal to s) recurs unless m − 1 has a common factor (greater than unity) with the quotient of sm by the highest common factor of sm with the sum of the digits of x.