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Operations on positive integers

Source: Mexican Math Olympiad 2012 - problem 4

December 1, 2013
inequalitiescombinatorics unsolvedcombinatorics

Problem Statement

The following process is applied to each positive integer: the sum of its digits is subtracted from the number, and the result is divided by 99. For example, the result of the process applied to 938938 is 102102, since 938−(9+3+8)9=102.\frac{938-(9 + 3 + 8)}{9} = 102. Applying the process twice to 938938 the result is 1111, applied three times the result is 11, and applying it four times the result is 00. When the process is applied one or more times to an integer nn, the result is eventually 00. The number obtained before obtaining 00 is called the house of nn.
How many integers less than 2600026000 share the same house as 20122012?
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