Problem

Source: 2019 Pan-African Shortlist - C2

Tags: number theory, combinatorics, sum of digits



On the board, we write the integers $1, 2, 3, \dots, 2019$. At each minute, we pick two numbers on the board $a$ and $b$, delete them, and write down the number $s(a + b)$ instead, where $s(n)$ denotes the sum of the digits of the integer $n$. Let $N$ be the last number on the board at the end. Is it possible to get $N = 19$? Is it possible to get $N = 15$?