Let $s(n)$ denote the sum of the digits of $n$. a) Do there exist distinct positive integers $a, b$, such that $2023a+s(a)=2023b+s(b)$? b) Do there exist distinct positive integers $a, b$, such that $a+2023s(a)=b+2023s(b)$?
Source: Italy MO 2023 P3
Tags: number theory
Let $s(n)$ denote the sum of the digits of $n$. a) Do there exist distinct positive integers $a, b$, such that $2023a+s(a)=2023b+s(b)$? b) Do there exist distinct positive integers $a, b$, such that $a+2023s(a)=b+2023s(b)$?