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4. Can 19 different positive integers be found, the sum of whose digits is equal, and whose sum is 1999?

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

I solved it I didn't Skip

Official solution

Hint: It cannot. Using property (1), we know these numbers are congruent to 1 modulo 9. Mimicking Example 3, we can enumerate to prove it.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.