Question 178
Let $a_{1}, a_{2}, a_{3}$ be positive integers. For every integer $n \geqslant 3$, set
$$
a_{n+1}=\operatorname{lcm}\left(a_{n}, a_{n-1}\right)-\operatorname{lcm}\left(a_{n-1}, a_{n-2}\right)
$$
it being understood that $\operatorname{lcm}(0, x)=0$ for every integer $x$.
Prove that there exists a natural number $k$ such that $k \leqslant a_{3}+4$ and $a_{k} \leqslant 0$.
$$
a_{n+1}=\operatorname{lcm}\left(a_{n}, a_{n-1}\right)-\operatorname{lcm}\left(a_{n-1}, a_{n-2}\right)
$$
it being understood that $\operatorname{lcm}(0, x)=0$ for every integer $x$.
Prove that there exists a natural number $k$ such that $k \leqslant a_{3}+4$ and $a_{k} \leqslant 0$.