Let \(N\) be the smallest positive integer such that the product of \(99\) and \(N\) is a number with all digits \(1\). What is the digit sum of \(N\)?

This problem is posed by Kiriti M.

**Details and assumptions**

The **digit sum** of a number is the sum of all its digits. For example the digit sum of 1123 is \(1 + 1 + 2 + 3 = 7\).

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