Summing Digits

Logic Level 4

It's common that, given a positive integer \(n\), to find the sum of its digits. For example, 2016 has digit sum 9, and 9686 has digit sum 29. This process can be iterated until we get a digit sum that has only a single digit. For example, 2016 arrives to the single-digit 9 in a single step, but 9686 takes three steps, going to 29, 11, before arriving at 2.

Let \(Z(n)\) be the number of digit-sum iterations needed from \(n\) to arrive to a single-digit result. For example, \(Z(2016) = 1\) and \(Z(9686) = 3\).

Let \(X(z)\) be the smallest positive integer \(n\) such that \(Z(n) = z\). For example, \(X(1) = 10\) and \(X(2) = 19\).

Find \(X(4562) \bmod 6\).


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