Problem

Source: Mexico National Olympaid 2019 P5

Tags: number theory, algebra



Let $a > b$ be relatively prime positive integers. A grashopper stands at point $0$ in a number line. Each minute, the grashopper jumps according to the following rules: If the current minute is a multiple of $a$ and not a multiple of $b$, it jumps $a$ units forward. If the current minute is a multiple of $b$ and not a multiple of $a$, it jumps $b$ units backward. If the current minute is both a multiple of $b$ and a multiple of $a$, it jumps $a - b$ units forward. If the current minute is neither a multiple of $a$ nor a multiple of $b$, it doesn't move. Find all positions on the number line that the grasshopper will eventually reach.