Reducing Numbers
Gabriel play with numbers with two or more digits. He replaces the 2 left-most digits of the number by their sum and repeats this process until he gets a single-digit number.
For example, starting with the number he gets:
How many 3-digits numbers will result in the number doing the process shown above?
Problem from Brazilian Math Olympiad of Public Schools.
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1 solution
The recursive process, applied to an integer, is equivalent to finding the original number modulo 9 . Therefore we need to find x that satisfy x ≡ 1 ( m o d 9 ) and are in the range 1 0 0 ≤ x ≤ 9 9 9 .
1 0 0 ≤ x ≤ 9 9 9 ⟹ 1 0 0 ≤ 9 k + 1 ≤ 9 9 9 ⟹ 1 1 ≤ k ≤ 1 1 0
so, there would be 1 1 0 − 1 1 + 1 = 1 0 0 of such three digit integers.