Problem

Source: INAMO 2023 P5 (OSN 2023)

Tags: number theory, greatest common divisor, least common multiple, Indonesia, Indonesia MO



Let $a$ and $b$ be positive integers such that $\text{gcd}(a, b) + \text{lcm}(a, b)$ is a multiple of $a+1$. If $b \le a$, show that $b$ is a perfect square.