Track / Stage 4 / 331 of 340 #591 of 1964

Problem 591

AMC 12 late, AIME early

Number theory Difficulty 5.0 Prove it

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.

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 reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.