Take a number $(x)$, then create the complete list of the numbers formed by deleting single digits from its base ten representation $(d_1,d_2,...,d_n)$. If the sum of those new numbers equals $x$ we call the number a special number.
Example :
1729404 = 729404 (delete 1) + 129404 (delete 7) + 179404 + 172404 + 172904 + 172944 + 172940
13758846 = 3758846 + 1758846 + 1358846 + 1378846 + 1375846 + 1375846 + 1375886 + 1375884
(in the 2nd example we see 1375846 twice.)
Question :
- Is the list of these kind of numbers infinite?
- If not, find the largest number (without the aid of a computer) with this property, then prove it!